globalbehaviorofsolutionsinapredator-preycross-diffusion...

Research ArticleGlobal Behavior of Solutions in a Predator-Prey Cross-DiffusionModel with Cannibalism

Meijun Chen Shengmao Fu and Xiaoli Yang

College of Mathematics and Statistics Northwest Normal University Lanzhou 730070 China

Correspondence should be addressed to Shengmao Fu fusmnwnueducn

Received 20 February 2020 Accepted 7 May 2020 Published 22 May 2020

Academic Editor Lucia Valentina Gambuzza

Copyright copy 2020 Meijun Chen et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

e global asymptotic behavior of solutions in a cross-diffusive predator-prey model with cannibalism is studied in this paperFirstly the local stability of nonnegative equilibria for the weakly coupled reaction-diffusion model and strongly coupled cross-diffusion model is discussed It is shown that the equilibria have the same stability properties for the corresponding ODE modeland semilinear reaction-diffusion model but under suitable conditions on reaction coefficients cross-diffusion-driven Turinginstability occurs Secondly the uniform boundedness and the global existence of solutions for the model with SKT-type cross-diffusion are investigated when the space dimension is one Finally the global stability of the positive equilibrium is established byconstructing a Lyapunov function e result indicates that under certain conditions on reaction coefficients the model has nononconstant positive steady state if the diffusion matrix is positive definite and the self-diffusion coefficients are large enough

1 Introduction

In 1999 Magnusson [1] proposed the following predator-prey model

dX

dt minus MmatX + AY + BXY + CXZ

dY

dt RX minus AY minus MjuvY minus SXY

dZ

dt TZ minus UZ

2minus VXZ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

where X Y and Z represent the biomasses of adult pred-ators juvenile predators and prey respectively Mmat andMjuv represent the death rates of adult predators and juv-enile predators respectively A denotes the specific rate ofjuveniles recruited to the adults class R denotes the birthrate T and U are logistic coefficients S represents thecannibalism attack rate B cS where c is the conversionefficiency of eaten juveniles into adult biomass and thebiological meanings of V and C are similar to S and B Formore details on model (1) please refer to [1]

Let

u S

MmatX

v SA

M2mat

Y

w C

MmatZ

1113957t Mmatt

(2)

and still denote 1113957t by t then (1) becomes

du

dt minus u + v + auv + uw

dv

dt bu minus gv minus uv

dw

dt sw minus rw

2minus euw

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(3)

HindawiComplexityVolume 2020 Article ID 1265798 19 pageshttpsdoiorg10115520201265798

where

a BMmat

SA

cMmat

A

b RA

M2mat

g A + Mjuv

Mmat

s T

Mmat

r U

C

e V

S

(4)

We can see that e is the only parameter depending on S

in system (3)Most works on cannibalism adopt a McKendrickndashvon

Foerster age-structured model [2ndash4] but the model isusually rewritten as an ODE system or nonlinear Volterraintegral equations [4 5] In 1995 Kohlmeier and Ebenhoh[6] studied a two-dimensional ODE model without anystructure and they obtained that in some cases cannibalismcan lead to a higher long term predator stock size en vanden Bosch and Gabriel [7] found that cannibalism canstabilize a predator-prey system in a structured model wherethe oscillations are due to age structure

Magnusson [1] discussed model (1) which is in someways a simplification of the system studied by van den Boschand Gabriel in [7] Magnusson made two simplificationsFirstly it is assumed that all juveniles are vulnerable topredation by the adults Secondly instantaneous maturationinto the adult class is proportional to the present juvenilebiomass ie a constant per capita rate of maturation and

making this simplifying assumption means that any oscil-lations that may occur are not caused by a delay inherent inthe system Moreover the stability and Hopf bifurcation ofsolutions for system (3) were studied in [1] e authorobtained the following conclusions On one hand if themortality rate of juveniles is low andor the recruitment rateto the mature population is high then there is a stableequilibrium with all three population sizes as positive Onthe other hand if the mortality rate of juveniles is high andor the recruitment rate to the mature population is low thenthe equilibrium will be stable for low levels of cannibalismbut a loss of stability by a Hopf bifurcation will take place asthe level of cannibalism increases

As we all know the growth of biological populationdepends not only on time but also on spatial distributionSpatial species interaction includes the self-diffusion whichis the natural dispersive force of movement of individualand the cross-diffusion that is the population fluxes of onespecies due to the presence of other species [8ndash16] ere-fore taking into account the effect of competition for re-sources on species growth law we are naturally led to thefollowing weakly coupled reaction-diffusion model

ut d1Δu minus u + v + auv + uw x isin Ω tgt 0

vt d2Δv + bu minus gv minus uv x isin Ω tgt 0

wt d3Δw + sw minus rw2 minus euw x isin Ω tgt 0

zηu zηv zηw 0 x isin zΩ tgt 0

u(x 0) u0(x)

v(x 0) v0(x)

w(x 0) w0(x)

x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)

and the Shigesada-Kawasaki-Teramoto model with stronglycoupled cross-diffusion

ut Δ d1u + α11u2 + α12uv + α13uw( 1113857 minus u + v + auv + uw x isin Ω tgt 0

vt Δ d2v + α21uv + α22v2 + α23vw( 1113857 + bu minus gv minus uv x isin Ω tgt 0

wt Δ d3w + α31uw + α32vw + α33w2( 1113857 + sw minus rw2 minus euw x isin Ω tgt 0


u(x 0) u0(x)

v(x 0) v0(x)

w(x 0) w0(x)

x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)

where Ω sub Rn is a bounded domain with smooth boundaryzΩ and η is the outward unit normal vector of the boundaryzΩ zη zzη e coefficients di(i 1 2 3) are the diffu-sion rates of u v andw αii(i 1 2 3) are referred as self-diffusion pressures and αij(ine j i j 1 2 3) are cross-diffusion coefficients which may be positive negative or

zero [17] e functions u0 v0 andw0 are nonnegativefunctions which are not identically zero

Very recently Zhang et al [18] found that the positiveequilibrium of (3) with r 0 can undergo stability switch(from stable to unstable to stable or from unstable to stableto unstable) with the change of the cannibalization rate

2 Complexity

eir results indicate that large cannibalization rate canmake the positive equilibrium globally stable although itsstability would change with the increase of the rate In factan important research subject for cannibalism models is thestabilizingdestabilizing effect of cannibalism [1 6 19] Forthis we need to compare the effect of cannibalism on thedynamic behavior for the ODE system the semilinear re-action-diffusion system and the quasi-linear cross-diffusionsystem

In this paper we first prove that stability properties ofequilibria for the ODE model (3) and the semilinear reac-tion-diffusion model (5) is similar but under suitableconditions on reaction coefficients cross-diffusion-drivenTuring instability in the quasi-linear model (6) occurs It isfound that the cross diffusion rate α13 is a decisive factor ofdestabilizing positive steady state that is cannibalism has nolonger a stabilizing effect en the uniform boundednessand the global existence of time-varying solutions for thecross-diffusion system (6) are investigated by using theenergy estimates and GagliardondashNirenberg-type inequalitieswhen the space dimension is one Finally some criteria onthe global asymptotic stability of the positive equilibriumpoint for (6) are given by Lyapunov function e obtainedresults indicate that for any cannibalism rate e undercertain conditions on the other reaction coefficients themodel has no nonconstant positive steady state if the dif-fusion matrix is positive definite and the self-diffusion co-efficients are large enough

Moreover we in [14] discussed another cross-diffusionmodel

ut minus d1Δu minus u + v + auv + uw x isin Ω tgt 0

vt minus Δ d2v +d4v

ε + u21113888 1113889 bu minus gv minus uv x isin Ω tgt 0

wt minus d3Δw sw minus rw2 minus euw x isin Ω tgt 0


u(x 0) u0(x)

v(x 0) v0(x)

w(x 0) w0(x)

x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)

where d4 is a cross-diffusion coefficient and ε is a positiveconstant (please refer to [12]) We proved that ifa b g s r e ε d1 d2 and d3 are fixed such that0lt εlt ε0 aglt 1 rgt s and (9) (see Section 2) hold then thereexists a positive constant dlowast4 such that (7) has at least onenonconstant positive steady state for d4 gedlowast4 is impliesthat the cross-diffusion rate is the decisive factor of

destabilization on the constant positive steady state andtherefore cannibalism is an auxiliary destabilizing forceRecently in another paper we detailedly describe the localstructure of the nonconstant steady states and discuss thestability and instability of the steady state bifurcation

2 Linearization Analysis

In this section according to the linearization analysis andRouthndashHurwitz criterion we focus on discussing the sta-bility of the reaction-diffusion system (5) and cross-diffusionsystem (6) To this end we first introduce the stability of theODE system (3)

21 ODE Model (3) Obviously problem (3) has trivialequilibrium E0(0 0 0) semitrivial equilibrium E1(0 0 sr)and E2((g minus b)(ab minus 1) (g minus b)(ag minus 1) 0) if glt blt 1aor ggt bgt 1a Moreover if r 0 bltg + ((s(1 minus ag))(e + as)) then a positive equilibrium point is given byE3(u v w) where

u s

e

v bs

eg + s

w e(g minus b) + s(1 minus ab)

eg + s

(8)

If rne 0 andbr + gs minus grgt 0

s + eg + r(1 minus ab)gt 0

s(1 minus ab) + e(g minus b)gt 0

⎧⎪⎪⎨

⎪⎪⎩(9)

then a positive equilibrium point is uniquely given by1113957u (1113957u 1113957v 1113957w) where

1113957u α +

α2 + 4eβ

1113968

2e

1113957v b1113957u

g + 1113957u

1113957w s minus e1113957u

r

α abr + s minus r minus eg

β br + gs minus gr

(10)

In fact the stability of these nonnegative equilibria in (3)has been obtained in [1] as follows

(1) E0 is unconditionally unstable(2) E1 is locally asymptotically stable if slt r and is

unstable if sgt r(3) E2 is locally asymptotically stable if glt blt 1a and is

unstable if bltg + ((s(1 minus ag))(e + as))

Complexity 3

(4)

(a) If aglt 1 then E3 is locally asymptotically stable(b) If aggt 1 S 0 (ie e⟶infin) and bltg then E3

is locally asymptotically stable(c) If aggt 1 Sne 0 (ie eneinfin) bltg and ablt 1 then

there exists elowast such that E3 is locally asymp-totically stable when egt elowast and is unstable whenelt elowast

Now we give the stability of 1113957u with respect to model (3)

Theorem 1 -e positive equilibrium 1113957u of (3) is locally as-ymptotically stable if aglt 1

Proof e Jacobian matrix of (3) to the generic us (us

vs ws) reads

J us( 1113857

j11 j12 j13

j21 j22 j23

j31 j32 j33

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (11)

wherej11 minus 1 + avs + ws

j12 1 + aus

j13 us

j21 b minus vs

j22 minus g minus us

j23 0

j31 minus ews

j32 0

j33 s minus 2rws minus eus

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

A direct calculation yields that the characteristic equa-tion of J(1113957u) is

φ(λ) |λI minus J| λ3 + α1λ2

+ α2λ + α3 0 (13)

with

α1 11113957u

g1113957u + 1113957v + +1113957u2

+ r1113957u1113957w1113872 1113873gt 0

α2 11113957u

(1 minus ag)1113957u1113957v + r1113957w(1113957v + g1113957u) +(e + r)1113957u2

1113957w1113872 1113873

α3 11113957u

e1113957w1113957u2(g + 1113957u) + r(1 minus ag)1113957u1113957v1113957w1113872 1113873

α1α2 minus α3 r1113957w1113957u2

+ (1 minus ag)1113957v + 2gr1113957w + re1113957w2

+ r2

1113957w2

1113872 11138731113957u

+ rg2

1113957w + e1113957v1113957w + 2r1113957v1113957w + gr2

1113957w2

+(1 minus ag)g1113957v +r1113957w1113957v2

1113957u2 +11113957u

middot (1 minus ag)1113957v2

+ r2

1113957w21113957v + 2gr1113957v1113957w1113872 1113873

(14)

It is easy to see that α3 gt 0 and α1α2 minus α3 gt 0 thanks toaglt 1erefore by RouthndashHurwitz criterion we know that(1113957u 1113957v 1113957w) is locally asymptotically stable

Notice that the unstable equilibrium points of (3) arealso unstable for (5) and (6) erefore for systems (5) and(6) we only discuss the stability of the equilibrium pointswhich are stable for (3)

22 Weakly Coupled Reaction-Diffusion System (5) Let 0

μ1 lt μ2 lt μ3 lt middot middot middot be the eigenvalues of the operator minus Δ on Ωwith the homogeneous Neumann boundary condition andlet E(μi) be the eigenspace corresponding to μi in H1(Ω)Let X be the closure of [C1(Ω)]3 in [H1(Ω)]3ϕij j 1 2 dimE(μi)1113966 1113967 be an orthonormal basis of

E(μi) and let Xij cϕij c isin R31113966 1113967 en

X oplus+infin

i1Xi

Xi oplusdimE μi( )

j1Xij

(15)

For system (5) by the linearization analysis and somesimilar arguments to the proof of eorem 2 andeorem 3in [12] we can obtain the following two theorems

Theorem 2

(1) -e semitrivial equilibrium E1 of (5) is locally as-ymptotically stable if slt r

(2) -e semitrivial equilibrium E2 of (5) is locally as-ymptotically stable if glt blt 1a

(3) -e positive equilibrium E3 of (5) is locally asymp-totically stable either aglt 1 or aggt 1 S 0 (iee⟶infin) and bltg

Proof We only prove (3) e proof of (1) and (2) is similarto (3) so we omit it here Let u (u v w)D

diag(d1 d2 d3) and L DΔ + J(E3) where j11 minus (be(eg + s))lt 0 j12 (e + as)egt 0 j13 ugt 0 j21 (beg(eg + s))gt 0 j22 minus (g + u)lt 0 j23 0 j31 minus ewlt 0

and j32 j33 0 e linearization of (5) at E3 is ut LuFor each ige 1 Xi is invariant under the operatorL and λ isan eigenvalue of L if and only if it is an eigenvalue of thematrix minus μiD + J(E3) for some ige 1 in which case there is an

eigenvector in Xi Denote the characteristic polynomial ofminus μiD + J(E3) by φi(λ) λ3 + Aiλ

2 + Biλ + Ci whereAi μi d1 + d2 + d3( 1113857 minus j11 + j22( 1113857

Bi μ2i d1d2 + d2d3 + d1d3( 1113857 minus μi d1j22 + d2j11 + d3j11 + d3j22( 1113857

+ j11j22 minus j12j21 minus j13j31

Ci μ3i d1d2d3 minus μ2i d1d3a22 + d2d3j11( 1113857

+ μi d3j11j22 minus d3j12j21 minus d2j13j31( 1113857 + j22j13j31

(16)

A straightforward computation gives Hi AiBi minus Ci

I1μ3i + I2μ2i + I3μi + I4 where

4 Complexity

I1 d21 d2 + d3( 1113857 + d

22 d1 + d3( 1113857 + d

23 d1 + d2( 1113857 + 2d1d2d3

I2 minus j11 d22 + d

231113872 1113873 minus j22 d

21 + d

231113872 1113873

minus 2 j11 + j22( 1113857 d1d2 + d2d3 + d1d3( 1113857

I3 j11 + j22( 1113857 d1j22 + d2j11 + d3j11 + d3j22( 1113857

+ d1 + d2( 1113857 j11j22 minus j12j21( 1113857 minus d1 + d3( 1113857j13j31

I4 minus j11 + j22( 1113857 j11j22 minus j12j21( 1113857 + j11j13j31

(17)

Obviously if j11j22 gt j12j21 (ie aglt 1) thenHi gt 0 Ai gt 0 andCi gt 0 It follows from the RouthndashHurwitzcriterion that the three roots λi1 λi2 and λi3 of φi(λ) 0 allhave negative real parts Moreover if S 0 thene VS⟶infin w⟶ 1 minus bg j11⟶ minus (bg) j12⟶1 j13⟶ 0 j21⟶ b j22⟶ minus g and j31⟶ minus e(1minus (bg)) erefore if bltg then Hi gt 0 Ai gt 0 andCi gt 0

We claim that for each ige 1 there exists a positiveconstant δ such that

Re λi11113864 1113865Re λi21113966 1113967Re λi31113966 1113967le minus δ ige 1 (18)

Let λ μiξ en φi(λ) μ3i ξ3

+ Aiμ2i ξ2

+ Biμiξ +

Ci ≜ 1113957φi(ξ) Since μi⟶infin as i⟶ infin it follows that

limi⟶infin

1113957φi(ξ)

μ3i1113896 1113897 ξ3 + d1 + d2 + d3( 1113857ξ2

+ d1d2 + d2d3 + d1d3( 1113857ξ + d1d2d3 ≜ 1113957φ(ξ)

(19)

Clearly 1113957φ(ξ) 0 has three negative roots minus d1 minus d2 andminus d3 Let dlowast min d1 d2 d31113864 1113865 By continuity one can see thatthere exists a i0 such that the three roots ξi1 ξi2 and ξi3 of1113957φi(ξ) 0 satisfy Re ξi11113864 1113865Re ξi21113864 1113865Re ξi31113864 1113865le minus dlowast2 when ige i0erefore if ige i0 then Re λi11113864 1113865Re λi21113864 1113865Re λi31113864 1113865leminus μidlowast2le minus dlowast2 Let 1113957δ minus max0leilei0 Re λi11113864 1113865Re λi21113864 11138651113864

Re λi31113864 1113865en 1113957δ gt 0 and (18) holds for δ min 1113957δ dlowast21113966 1113967 Sothe positive equilibrium point E3 of (5) is locally asymp-totically stable

In order to obtain the global asymptotic behavior of thesolutions to (5) or (6) we need the following result whichcan be found in [12]

Lemma 1 Let a and b be positive constants Assume thatφψ isin C1[a +infin) ψ(t)ge 0 and φ is bounded from below Ifφprime(t)le minus bψ(t) and ψprime(t) is bounded from above in [a +infin)then limt⟶infinψ(t) 0

Theorem 3

(1) -e positive equilibrium E3 of (5) is globally as-ymptotically stable if aglt 1 andeg + agslt egslt eg + s

(2) -e semitrivial equilibrium E2 of (5) is globally as-ymptotically stable if 1lt (1 minus ab)(b minus g)lt bg

Proof Let (u v w) be the unique positive solution of (5)Estimates similar to eorem 21 in [15] and eorem

A2 in [20] show

u(middot t)C2+α(Ω)

v(middot t)C2+α(Ω)

u(middot t)C2+α(Ω)leC forallt ge 1

(20)

where α isin (0 1) and C does not depend on t

(1) Define

V(u v w) 1113946Ω

u minus u minus u lnu

u1113874 1113875 + λ v minus v minus v ln

v

v1113874 11138751113876

+ ρ w minus w minus w lnw

w1113874 11138751113877dx

(21)

where λ (e + as)eg and ρ 1e ObviouslyV(u v w) is nonnegative and V(u v w) 0 if andonly if u u v v andw w e time derivativeof V(u v w) for (5) satisfies that

dV(u v w)

dt minus 1113946Ω

d1u

u2 |nablau|2

+λd2v

v2|nablav|

2+ρd3w

w2 |nablaw|2

1113888 1113889dx

minus1s

1113946Ω

(es minus e minus as)

s

v

u(u minus u)

2+

(e + as)(eg + s minus egs)

eg

u

v(v minus v)

2+

(e + as)

s

v

u

1113971

(u minus u) minus

s(e + as)u

v

1113970

(v minus v)⎛⎝ ⎞⎠

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭dx

le minus 1113946Ω

d1u

u2 |nablau|2

+λd2v

v2|nablav|

2+ρd3w

w2 |nablaw|2

1113888 1113889dx

minus1s

1113946Ω


s

v

u(u minus u)

2+


eg

u

v(v minus v)

21113896 1113897dx

(22)

Complexity 5

From eorem 2 Lemma 1 and (20) we know that

limt⟶infin

1113946Ω

|nablau|2

+|nablav|2

+|nablaw|2

1113872 1113873dx 0 (23)

limt⟶infin

1113946Ω

(u minus u)2dx 0

limt⟶infin

1113946Ω

(v minus v)2dx 0

(24)

It follows from the Poincare inequality that

limt⟶infin

1113946Ω

(u minus 1113954u)2dx 0

limt⟶infin

1113946Ω

(v minus 1113954v)2dx 0

limt⟶infin

1113946Ω

(w minus 1113954w)2dx 0

(25)

where 1113954f 111393810 fdx As |1113954u minus u|2 1113938Ω (1113954u minus u)2dxle

21113938Ω(1113954u minus u)2dx + 21113938Ω(u minus u)2dx from (24) and(25) we have

limt⟶infin

1113954u(t) u

limt⟶infin

1113954v(t) v(26)

erefore there exists a sequence tm1113864 1113865 such that1113954uprime(tm)⟶ 0 as tm⟶infin Since 1113954w(tm)1113864 1113865 isbounded so there exists a subsequence still denotedby 1113954w(tm)1113864 1113865 such that 1113954w(tm)⟶ 1113957w as tm⟶infinFrom the first equation of (5) we have

1113954uprime tm( 1113857 1113946Ω

[(av + w minus 1)(u minus u)

+(au + 1)(v minus v) + u(w minus w)]dx|tm

(27)

Let m⟶infin in the above equation and from (25)we have

limtm⟶infin

1113954w(t) w (28)

On the contrary (20) implies that there exists asubsequence still denoted by tm1113864 1113865 and nonnegativefunctions ulowast vlowast wlowast isin C2(Ω) such that

u middot tm( 1113857 minus ulowastC2(Ω)⟶ 0

v middot tm( 1113857 minus vlowastC2(Ω)⟶ 0

w middot tm( 1113857 minus wlowastC2(Ω)⟶ 0

m⟶infin

(29)

Combining this with (25)ndash(28) one can obtain thatulowast u vlowast v wlowast w and

u middot tm( 1113857 minus u

C2(Ω)⟶ 0

v middot tm( 1113857 minus v

C2(Ω)⟶ 0

w middot tm( 1113857 minus w

C2(Ω)⟶ 0

m⟶infin

(30)

e global asymptotic stability of E3 follows fromthis together with eorem 2

(2) Define

V(u v w) 1113946Ω

u minus u2 minus u2 lnu

u21113888 11138891113890

+ l v minus v2 minus v2 lnv

v21113888 1113889 +

1e

w1113891dx

(31)

where l (1 minus ag)(g(1 minus ab)) and u2 (g minus b)(ab minus 1) v2 (g minus b)(ag minus 1) en

dV(u v w)

dt minus 1113946Ω

d1u2

u2 |nablau|2

+λd2v2

v2|nablav|

21113888 1113889dx minus 1113946

Ω

(1 minus ab)(b minus g) minus 1(b minus g)2

v

uu minus u2( 1113857

21113896

+(1 minus ag)2 b

g(b minus g)(1 minus ab)minus 11113888 1113889

u

vv minus v2( 1113857

2+

r

ew

2

+1

b minus g

v

u

1113970

u minus u2( 1113857 minus (1 minus ag)

u

v

1113970

v minus v2( 11138571113888 1113889

2⎫⎬

⎭dx

(32)

From this (2) can be proved using similar argumentsas in (1)

Remark 1 e stability of 1113957u is demonstrated specifically in[14] that is 1113957u is locally asymptotically stable if aglt 1

23 Strongly Coupled Cross-Diffusion System (6)Comparing Sections 21 and 22 we find that Ei i 1 2 3and 1113957u have the same stability properties in systems (3) and(5) For the sake of convenience we denote nonnegativeequilibria Ei i 1 2 3 and 1113957u by us (us vs ws) Now we

6 Complexity

show that the destabilization effect of cross-diffusion onus (us vs ws)

Linearizing system (6) at an equilibrium (us vs ws) wecan obtain

zuzt

(D + P)Δu + Ju x isin Ω tgt 0

zuz]

0 x isin zΩ tgt 0

u(x 0) u0(x) x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(33)

where u (u v w)T D diag(d1 d2 d3) and J is given in(11)

P

p11 p12 p13

p21 p22 p23

p31 p33 p33

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (34)

with

p11 2α11us + α12vs + α13ws

p12 α12us

p13 α13us

p21 α21vs

p22 α21us + 2α22vs + α23ws

p23 α23vs

p31 α31ws

p32 α32ws

p33 α31us + α32vs + 2α33ws

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(35)

We denote

L ≔ minus μi(D + P) + J (36)

then the corresponding characteristic polynomial of L is

ϕ λ3 + c1λ2

+ c2λ + c3 (37)

where

c1 (trD + trP)μi minus trJ

c2 β1μ2i + β2μi + β3

c3 det μiP minus J( 1113857 β4μ3i + β5μ

2i + β6μi + β7

(38)

with

β1 d1d2 + d1d3 + d2d3 + d1 p22 + p33( 1113857 + d2 p11 + p33( 1113857 + d3 p11 + p22( 1113857 + M11 + M22 + M33

β2 minus j11 d2 + d3 + p22 + p33( 1113857 minus j22 d1 + d3 + p11 + p33( 1113857 minus j33 d1 + d2 + p11 + p22( 1113857 + p21j12 + p12j21 + p31j13 + p13j31

β3 M22 + M33 + j22j33

β4 det(D) + det(P) + d1 d2p33 + d3p22 + M11( 1113857 + d2M22 + d3M33 + d2d3P11

β5 minus j11 M11 + d2p33 + d3p22 + d2d3( 1113857 + j12 M12 + d3p21( 1113857 + j13 d2p31 minus M13( 1113857

+ j21 d3p12 + M21( 1113857 minus j22 M22 + d1p33 + d3p11 + d1d3( 1113857

+ j31 d2p13 minus M31( 1113857 minus j33 M33 + d1d2 + d1p22 + d2p11( 1113857

β6 p11 + d1( 1113857j22j33 minus p12j21j33 minus p13j22j31 minus p21j12j33 + p22j11j33 minus p22j13j31

+ p23j12j31 minus p31j13j22 + p32j13j21 + p33 + d3( 1113857M33 + d2M22

β7 minus j33M33 + j13j22j31

(39)

where Mij and Mij i j 1 2 3 are cofactors of matrix P

and J respectivelyerefore according to the principle of the linearized

stability ((21] 86) (22] 52)) the local stability ofnonnegative equilibria of model (6) is given below

Theorem 4 Let di and αij i j 1 2 3 be positive constants-en the following statements for system (6) hold

(1) us is locally asymptotically stable if and only if forevery i isin N all the eigenvalues of the linearizationmatrix L have negative real part

(2) us is unstable if and only if there exists an i isin N suchthat the linearization matrix L has at least one ei-genvalue with positive real part

By applying the RouthndashHurwitz criterion or Corollary 22in [23] we have the following stability and instability results

Corollary 1

(1) E1 is locally asymptotically stable if slt r and bltgmhere m min s (r minus s)r rs2(r minus s)

(2) E2 is locally asymptotically stable if glt blt 1a(1 minus ab)(1 minus ag)lt slt e(b minus g))(1 minus ab) and

Complexity 7

(ab minus 1)(ag minus 1)d3 + (ab minus 1)(g minus b)α32 + (g minus b)

(ag minus 1)α31 gt 0

(3) E3 is locally asymptotically stable if aglt 1 eα13 lt α31and (g + u)(2wα33 + d3) + uvα32 gt s(uα21 + 2vα22 +

wα23 + d2)

(4) 1113957u is locally asymptotically stable if slt δ ≔ r(1 minus ab)minus

eg +

(abr minus r + eg)2 + 4er(g minus b)

1113969

bltg aglt 1and (er)α23 le α21 le α13 le (α31e)le (α21α33eα23)

(5) If aglt 1 and sgtf then there exists a positive con-stant 1113957α13 such that 1113957u is unstable when α13 gt 1113957α13

(6) If aggt 1 then there exists a positive constant 1113957d3 suchthat 1113957u is unstable when d3 gt 1113957d3

Proof We only prove the case of (4) and (5) other cases canbe treated similarly With the help of the Mapleapplication we can evaluate that ci gt 0 i 1 2 3 and c1c2 minus

c3 gt 0 due to bltg aglt 1 slt δ ≔ r(1 minus ab) minus eg+


1113969

and (er)α23 lt α21 lt α13 lt(α31e)lt (α21α33eα23) erefore RouthndashHurwitz criterionshows that 1113957u is locally asymptotically stable

On the contrary the constant equilibrium is unstable ifand only if there exists an i isin N such that c3 lt 0 or c1c2 lt c3anks to calculate c1c2 minus c3 is complicated we now onlydiscuss the case of c3 lt 0

In fact c3 det(μiP minus J) β4μ3i + β5μ2i + β6μi+

β7 ≔ C(α13 μi) where β4 gt 0 In order to study whethercross-diffusion has unstable effect we assume that 1113957u is stablein the corresponding self-diffusion system ie aglt 1 whichimplies that β7 minus j331113957v(1 minus ag) + j13j22j31 gt 0 Let 1113957μ1 1113957μ2and 1113957μ3 be the three roots of C(α13 μi) with Re 1113957μ11113864 1113865leRe 1113957μ21113864 1113865leRe 1113957μ31113864 1113865 en 1113957μ11113957μ21113957μ3 minus (β4β1)lt 0 at least one of1113957μ1 1113957μ2 and 1113957μ3 is real and negative and the product of theother two is positive

Now we consider the following limits

C1 ≔ limα13⟶infin

β1α13

21113957w3α23α33 + 21113957uα21α33 + 41113957vα22α33 + 2d3α33 + d3α23( 1113857 1113957w

2

+ 21113957vα32 1113957uα21 + 1113957vα22( 1113857 + d2 d3 + 1113957vα32( 1113857 + d3 1113957uα21 + 21113957vα22( 1113857


β2α13

(r1113957w minus e1113957u) 1113957u1113957wα21 + 21113957v1113957wα22 + 1113957w2α231113872 1113873

+(21113957u1113957v + g1113957v minus b1113957u)1113957wα32 + 21113957w2(g + u)α33


β3α13

(s minus 2e1113957u)(g + 1113957u)

(40)

Note that

limα13⟶infin

C α13 ]i( 1113857

α13 μi C1μ

2i + C2μi + C31113872 1113873 (41)

and C1 gt 0 C3 lt 0 since slt 2e1113957u ie sgt r(1 minus ab)minus

eg +


1113969

A continuity argumentyields that 1113957μ1 is real and negative Moreover as 1113957μ21113957μ3 gt 0 1113957μ2and 1113957μ3 are real and positive and

limα13⟶infin

1113957μ1 minus C2 minus

C2 minus 4C1C3

1113968

2C1lt 0

limα13⟶infin

1113957μ2 0

limα13⟶infin

1113957μ3 minus C2 +

C2 minus 4C1C3

1113968

2C1gt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(42)

us there exists a positive constant 1113957α13 such that for allα13 gt 1113957α13 the three roots 1113957μ1 1113957μ2 and 1113957μ3 of C(α13 μi) are allreal and satisfy

minus infinlt 1113957μ1 lt 0lt 1113957μ2 lt 1113957μ3

α3 C α13 μi( 1113857lt 0 when μi isin minus infin 1113957μ1( 1113857cup 1113957μ2 1113957μ3( 1113857

α3 C α13 μi( 1113857gt 0 when μi isin 1113957μ1 1113957μ2( 1113857cup 1113957μ3 +infin( 1113857

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(43)

erefore the characteristic equation ϕ 0 has at leastone positive eigenvalue when μi isin (1113957μ2 1113957μ3)

Remark 2 Remark 1 and conclusion (5) of Corollary 1indicate that if aglt 1 and

sgtf the large cross-diffusion

rate α13 has destabilizing effect for positive equilibrium point(1113957u 1113957v 1113957w) is means that under suitable conditions onreaction coefficients cross-diffusion-driven Turing insta-bility occurs if the cross-diffusion rate is sufficiently large

To demonstrate the results of stability and instability for1113957u we give the following examples

Example 1 Assume that the spatial domain Ω [0 10π]

(1) Let

8 Complexity

α11 α12 α21 α22 α23 α32 1

α13 12

α31 3

α33 4

a 05

b 1

g 15

s 3

e 2

r 4

d1 3

d2 3

d3 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(44)

en the equilibrium point (1113957u 1113957v 1113957w) (0725

0326 0388) c1 5809gt 0 c2 10163gt 0 c3

5082gt 0 and c1c2 minus c3 53959gt 0 From Corol-lary 1 (4) 1113957u is locally asymptotically stable

(2) Let

α11 α12 α21 α22 α23 α31 α32 α33 1

d1 3

d2 3

d3 2

a 05

b 13

g 15

s 3

e 5

r 4

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(45)

and choose α13 300 en the equilibrium point(1113957u 1113957v 1113957w) (0453 0302 0184) ag 075lt 1 s

3gtf 1195 and c3 minus 0663lt 0 erefore 1113957u isunstable thanks to Corollary 1 (5)

(3) Let αij 01 i j 1 2 3 a 28 b 1 g 3 s

3 e 3 r 45 d1 3 andd2 3 and choosed3 200 en the equilibrium point (1113957u 1113957v 1113957w)

(07 0189 02) ag 84gt 1 and c3 minus 1643lt 0us 1113957u is unstable due to Corollary 1 (6)

3 The Uniform Boundedness and the GlobalExistence of Solutions for (6)

In this section the uniform boundedness and the globalexistence of solutions for (6) are investigated when the spacedimension is one For simplicity let Ω (0 1) denote| middot |kp middot Wk

p(01) | middot |p middot Lp(01) andQT Ω times (0 T)From the consequence of a series of important papers[24ndash26] by Amann we have if u0 v0 w0 isinW1

p(Ω) withpgt 1 then (6) has a unique nonnegative solutionu v w isin C([0 T) W1

p(Ω))capCinfin((0 T) Cinfin(Ω)) whereTle +infin is the maximal existence time for the local solutionIf the solution (u v w) satisfies the estimates

sup u(middot t)w1p(Ω) v(middot t)w1

p(Ω) w(middot t)w1p(Ω) 0lt tltT1113882 1113883ltinfin

(46)

then T +infin Moreover if u0 v0 w0 isinW2p(0 1) then

u v w isin C([0infin) W2p(0 1))

In order to establish the timing uniform W12minus estimate of

the solution for system (6) the following corollaries to theGagliardondashNirenberg-type inequality play important roles

Lemma 2 -ere exists a universal constant C such that

|u|2 leC ux

11138681113868111386811138681113868111386811138681113868132 |u|

231 + |u|11113874 1113875 forallu isinW

12(0 1) (47)

|u|4 leC ux

11138681113868111386811138681113868111386811138681113868122 |u|

121 + |u|11113874 1113875 forallu isinW

12(0 1) (48)

|u|52 leC ux

11138681113868111386811138681113868111386811138681113868252 |u|

351 + |u|11113874 1113875 forallu isinW

12(0 1) (49)

ux

111386811138681113868111386811138681113868111386811138682leC uxx

11138681113868111386811138681113868111386811138681113868352 |u|

251 + |u|11113874 1113875 forallu isinW

22(0 1) (50)

Theorem 5 Let u0 v0 w0 isinW22(0 1) and (u v w) be the

unique nonnegative solution for (6) in the maximal existenceinterval [0 T) Assume that

(H1) ablt 1 bltg

(H2)8α11α21α31 gt α21α213 + α212α318α12α22α32 gt α32α221 + α223α128α13α23α33 gt α23α231 + α232α13

en there exist t0 gt 0 and positive constants M Mprimewhich depend only on di αij(i j 1 2 3) a b g s r e suchthat

sup |u(middot t)|12 |(v(middot t))|12 |w(middot t)|12 t isin t0 T( 11138571113966 1113967leMprime

(51)

max u(x t) v(x t) w(x t) (x t) isin [0 1] times t0 T1113858 11138591113864 1113865leM

(52)

and T infin Moreover if di ge 1(i 1 2 3) andd2d1 d3d1 isin [ d d] where d and d are positive constantsthen Mprime andM depend on d andd but do not ond1 d2 andd3

Complexity 9

In this section we always denote that C is Sobolevembedding constant or other kind of universal constantAj Bj andCj are some positive constants which dependonly on αij(i j 1 2 3) a b g s r e and Kj are positiveconstants depending on di and αij(i j 1 2 3)

a b g s r e When d1 d2 d3 ge 1 d1d2 d3d2 isin [d d] Kj

depend on d andd but do not on d1 d2 and d3

Proof By the maximum principle one can obtain that thesolutions of (6) with nonnegative initial values are alwaysnonnegative We will give the W1

2-estimates of the solution(u v w) for (6) next

Firstly we establish L1-estimates of the solution (u v w)

for (6) Taking integrations of the first three equations in (6)over the domain [0 1] respectively and then combining thethree integration equalities linearly we have

ddt

11139461

0u + ρv +

1e

w1113874 1113875dxleC1 minus l 11139461

0u + ρv +

1e

w1113874 1113875dx

(53)

where C1 (s2er)|Ω| ρ (12)(max a 1g1113864 1113865 + (1b)) andl min 1 minus ρb ((ρg minus 1)ρ) s1113864 1113865 It follows from (H1) thatthere exists a positive constant τ0 such that

11139461

0udx 1113946

1

0vdx 1113946

1

0wdxleM0 tgt τ0 (54)

where M0 (C1l)max 1 (1ρ) e1113864 1113865 Moreover there exists apositive constant M0prime which depends on a b g s r e and theL1minus norm of u0 v0 and w0 such that

11139461

0udx 1113946

1

0vdx 1113946

1

0wdxleM0prime tge 0 (55)

Secondly we will obtain L2-estimates of (u v w) Wemultiply the first three equations in (6) by u v andw re-spectively and integrate over [0 1] to have

12ddt

11139461

0u2dxle minus d1 1113946

1

0u2xdx minus 1113946

1

0p11u

2xdx minus 1113946

1

0p12uxvxdx minus 1113946

1

0p13uxwxdx + 1113946

1

0uv + au

2v + u

2w1113872 1113873dx

12ddt

11139461

0v2dxle minus d2 1113946

1

0v2xdx minus 1113946

1

0p22v

2xdx minus 1113946

1


1

0p23vxwxdx + b 1113946

1

0uvdx

12ddt

11139461

0w

2dxle minus d3 11139461

0w

2xdx minus 1113946

1

0p33w

2xdx minus 1113946

1

0p31uxwxdx minus 1113946

1

0p32vxwxdx + s 1113946

1

0w

2dx

(56)

where pij i j 1 2 3 are given by (35) for (u v w) Letdlowast min d1 d2 d31113864 1113865 en we obtain that

12ddt

11139461

0u2

+ v2

+ w2

1113872 1113873dxle minus dlowast

11139461

0u2x + v

2x + w

2x1113872 1113873dx minus 1113946

1

0q ux vx wx( 1113857dx

+(b + 1) 11139461

0uvdx + a 1113946

1

0u2vdx + s 1113946

1

0w

2dx

(57)

where

q ux vx wx( 1113857 p11u2x + p22v

2x + p33w

2x + p12 + p21( 1113857uxvx

+ p23 + p32( 1113857vxwx + p13 + p31( 1113857uxwx

(58)

is a positive semidefinite quadratic form of ux vx andwx if(H2) holds So (H2) implies that

12ddt

11139461

0u2

+ v2

+ w2


11139461

0u2x + v

2x + w

2x1113872 1113873dx

+(b + 1) 11139461

0uvdx

+ a 11139461

0u2vdx + s 1113946

1

0w

2dx

(59)

10 Complexity

Now we proceed in the following two cases

(1) tge τ0 Notice by (48) that |u|44 leC(|ux|22|u|21+

|u|41)leCM20(|ux|22 + M2

0) en by the Younginequality

a 11139461

0u2vdxle a 1113946

1

0

u4

2εdx + a 1113946

1

0

ε2v2dx

a2CM2

02dlowast

11139461

0v2dx +

dlowast

211139461

0u2xdx +

dlowastM20

2

(60)

Substituting (60) into (59) we have12ddt

11139461

0u2

+ v2

+ w2

1113872 1113873dxle minusdlowast

211139461

0u2x + v

2x + w

2x1113872 1113873dx

+ C2 11139461

0u2

+ v2

+ w2

1113872 1113873dx +dlowastM2

02

(61)

Inequality (47) implies that |u|62 leC|ux|22|u|41+

|u|61 leCM40(|ux|22 + M2

0) erefore

minus 11139461

0u2x + v

2x + w

2x1113872 1113873dxle 3M

20 minus

19CM2

011139461

0u2

+ v2

+ w2

1113872 1113873dx1113890 1113891

3

(62)

which together with (59) infers that

12ddt

11139461

0u2

+ v2

+ w2

1113872 1113873dxle minus C3dlowast

11139461

0u2

+ v2

+ w2

1113872 1113873dx1113890 1113891

3

+ C2 11139461

0u2

+ v2

+ w2

1113872 1113873dx + 2dlowastM

20

(63)

is means that there exist positive constants τ1 ge τ0and M1 depending on a b g s r e di(i 1 2 3)

such that

11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w

2dxleM1 tge τ1 (64)

When dlowast ge 1 M1 is independent of dlowast since the zeropoint of the right-hand side in (63) can be estimatedby positive constants independent of dlowast

(2) tge 0 Replacing M0 with M0prime and repeating estimates(60)ndash(63) one can obtain a new inequality which issimilar to (64) e coefficients of this new inequalitydepend not only on di(i 1 2 3) a b g s r e butalso on initial functions u0 v0 w0 en there existsa positive constant M1prime depending on a b g s r e

di(i 1 2 3) and the L2-norm of u0 v0 andw0 suchthat

11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w

2dxleM1prime tge 0 (65)

Finally L2minus estimates of ux vx and wx will be obtainedWe introduce the following scaling

1113957u u

d2

1113957v v

d2

1113957w w

d2

1113957t d1t

(66)

Denoting ξ d2d1 and η d3d1 and using u v w and t

instead of 1113957u 1113957v 1113957w and1113957t respectively then system (6) reduces tout Pxx + f(u v w) 0ltxlt 1 tgt 0

vt Qxx + k(u v w) 0ltxlt 1 tgt 0

wt Rxx + h(u v w) 0lt xlt 1 tgt 0

ux(x t) vx(x t) wx(x t) 0 x 0 1 tgt 0

u(x 0) u0(x)

v(x 0) v0(x)

w(x 0) w0(x)

0ltxlt 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(67)

where P u + α11ξu2 + α12ξuv + α13ξuw Q ξv + α21ξuv +

α22ξv2 + α23ξvw R ηw + α31ξuw + α32ξvw + α33ξw2 andf(u v w) minus dminus 1

1 u + dminus 11 v + aξuv + ξuw k(u v w) bdminus 1

1u minus gdminus 1

1 v minus ξuv h(u v w) w(sdminus 11 minus rξw minus eξu)

We still divide the subsequent discuss into two cases

(1) tge τlowast1( d1τ1) (namely tge τ1 in the original scale) Itis clear that

11139461

0udx 1113946

1

0vdx 1113946

1

0wdxleM0d

minus 12

11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w

2dxleM1dminus 22

|P|1 |Q|1 |R|1 leDK1dminus 12

(68)

Multiplying the first three equations in (67) byPt Qt andRt and integrating them over the domain[0 1] respectively then adding up the three inte-gration equalities we have12yprime(t) minus 1113946

1

0u2tdx minus ξ1113946

1

0v2tdx

minus η11139461

0w

2tdx minus ξ1113946

1

0q utvtwt( 1113857dx

+ 11139461

01+ p11ξ( 1113857utf + p12ξvtf + p13ξwtf1113858 1113859dx

+ ξ11139461

0p21utk + 1+ p22( 1113857vtk + p23wtk1113858 1113859dx

+ 11139461

0p31ξuth + p32ξvth + η+ p33ξ( 1113857wth1113858 1113859dx

(69)

Complexity 11

where y 111393810(P2

x + Q2x + R2

x)dx q is given by (58) Itis not hard to verify by (H2) that there exists apositive constant C4 depending only on αij(i j

1 2 3) such that

q ut vt wt( 1113857geC4(u + v + w) u2t + v

2t + w

2t1113872 1113873 (70)

erefore

12yprime(t)le minus 1113946

1

0u2tdx minus ξ 1113946

1

0v2tdx minus η1113946

1

0w

2tdx minus C4ξ 1113946

1

0(u + v + w) u

2t + v

2t + w

2t1113872 1113873dx

+ 11139461

01 + p11ξ( 1113857utfdx + 1113946

1

0ξ 1 + p22( 1113857vtkdx + 1113946

1

0η + p33ξ( 1113857wthdx

+ 11139461

0p12ξvtfdx + 1113946

1

0p13ξwtfdx + 1113946

1

0p21ξutkdx

+ 11139461

0p23ξwtkdx + 1113946

1

0p31ξuthdx + 1113946

1

0p32ξvthdx

(71)

By the Hӧlder inequality one can obtain the fol-lowing estimates

11139461

0u4dxleM

131 d

minus (23)1 1113946

1

0u5dx1113888 1113889

23

11139461

0u2v2dxleM

131 d

minus (23)1 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

11139461

0u3dxleM

231 d

minus (43)1 1113946

1

0u5dx1113888 1113889

13

11139461

0uv

2dxleM231 d

minus (43)1 1113946

1

0v5dx1113888 1113889

13

11139461

0u3wdxleM

131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

12

11139461

0w

5dx1113888 1113889

16

11139461

0uvwdxleM

231 d

minus (43)2 1113946

1

0u5dx1113888 1113889

16

11139461

0v5dx1113888 1113889

16

11139461

0u2vwdxleM

131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

16

(72)

Applying the Young inequality the Hӧlder in-equality the GagliardondashNirenberg inequality and

estimate (68) one can obtain the following estimatesfor the terms on the right-hand side of (71)

12 Complexity

minus 11139461

0u2tdxle minus

12

11139461

0P2xxdx + 1113946

1

0f2dx

minus ξ 11139461

0v2tdxle minus

ξ2

11139461

0Q

2xxdx + ξ 1113946

1

0k2dx

minus η11139461

0w

2tdxle minus

η2

11139461

0R2xxdx + η1113946

1

0h2dx

11139461

0f2dxle 2M1d

minus 21 d

minus 22 + a

2ξ2M131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

+ ξ2M131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0w

5dx1113888 1113889

13

+ 2aξdminus 11 d

minus (43)2 M

231 1113946

1

0v5dx1113888 1113889

13

+ 2ξdminus 11 d

minus (23)2 M

231 1113946

1

0u5dx1113888 1113889

16

11139461

0v5dx1113888 1113889

16

+ 2aξ2dminus (23)2 M

131 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

16

ξ 11139461

0k2dxle ξd

minus 21 d

minus 22 M1 b

2+ g

21113872 1113873 + ξ3dminus (23)

2 M131 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

+ ξ2gdminus 11 d

minus (43)2 M

231 1113946

1

0v5dx1113888 1113889

13

η11139461

0h2dxle ηs

2d

minus 21 d

minus 22 M1 + ηd

2ξ2dminus (23)2 M

131 1113946

1

0u5dx1113888 1113889

13

ηr2ξ2dminus (23)

2 M131

+ ηr2ξ2dminus (23)

2 M131 1113946

1

0u5dx1113888 1113889

23

+ η dr ξ2dminus (23)2 M

131 1113946

1

0u5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

12

(73)

us

minus 11139461


1


1

0w

2tdxle minus

12

11139461

0P2xxdx minus

ξ2

11139461

0Q

2xxdx minus

η2

11139461

0R2xxdx + C5(1 + ξ + η)d

minus 21 d

minus 22 M1

+ C6dminus (43)2 M

231 ξ(1 + η) 1113946

1

0u5

+ v5

1113872 1113873dx1113888 1113889

13

+ C7dminus (23)2 M

131 ξ2(1 + η) 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx1113888 1113889

23

(74)

Complexity 13

11139461

0utfdxle d

minus 11 1113946

1

0uutdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ d

minus 11 1113946

1

0utvdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ aξ 1113946

1

0uutvdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ ξ 1113946

1

0uutwdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le dminus 11

12ϵ

11139461

0udx +ϵ2

11139461

0uu

2tdx1113888 1113889 + d

minus 11

12ϵ

11139461

0vdx +ϵ2

11139461

0vu

2tdx1113888 1113889

+ aξ12ϵ

11139461

0uv

2dx +ϵ2

11139461

0uu

2tdx1113888 1113889 + ξ

12ϵ

11139461

0uw

2dx +ϵ2

11139461

0uu

2tdx1113888 1113889

ledminus 11 dminus 1

2ϵ

M0 +a

2ϵM

231 d

minus (43)2 ξ 1113946

1

0v5dx1113888 1113889

13

+ξ2ϵ

M231 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+ϵdminus 1

12

11139461

0vu

2tdx +ϵ2

dminus 11 + aξ + ξ1113872 1113873 1113946

1

0uu

2tdx

2α11ξ 11139461

0uutfdxle

α11ϵ

M231 d

minus 11 d

232 1113946

1

0u5dx1113888 1113889

13

+ 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦ +

3α11(a + 1)

5ϵξ2 1113946

1

0u5dx

+2α11a5ϵ

ξ2 11139461

0v5dx +

2α115ϵ

ξ2 11139461

0w

5dx + α11ξϵ 2dminus 11 + aξ + ξ1113872 1113873 1113946

1

0uu

2tdx

2α12ξ 11139461

0vutfdxle

α12ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0v5dx1113888 1113889

13

+α12(a + 1)

5ϵξ2 1113946

1

0u5dx +

2α125ϵ

ξ2 11139461

0w

5dx +α12(3a + 4)

10ϵξ2 1113946

1

0v5dx

α13ξ 11139461

0wutfdxle

α13ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α13(a +(12))

5ϵξ2 1113946

1

0u5dx +

2aα135ϵ

ξ2 11139461

0v5dx

+2α13(a + 1)

5ϵξ2 1113946

1

0w

5dx +α13dminus 1

12ϵξ 1113946

1

0vu

2tdx +ϵ2

α13ξdminus 11 + α13ξ

2+ 11113872 1113873 1113946

1

0uu

2tdx

ξ 11139461

0vtkdxle

b + g

2ϵM0d

minus 11 d

minus 12 ξ +

ξ2

2ϵM

231 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+bdminus 1

12ϵξ 1113946

1

0uv

2tdx +ϵ2ξ gd

minus 11 + ξ1113872 1113873 1113946

1

0vv

2tdx

α21ξ 11139461

0uvtkdxle

α21(b + g)

2ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+2α215ϵ

ξ2 11139461

0u5dx

+α2110ϵ

ξ2 11139461

0v5dx +

α21bdminus 11

2ϵξ 1113946

1

0uv

2tdx +

α21ϵξ2

gdminus 11 + ξ1113872 1113873 1113946

1

0vv

2tdx

2α22ξ 11139461

0vvtk dx le

α22(b + g)

ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+α225ϵ

ξ2 11139461

0u5dx

+4α225ϵ

ξ2 11139461

0v5dx + α22(b + g)d

minus 11 ϵξ 1113946

1

0vv

2tdx + α22ϵξ

211139461

0uv

2tdx

α23ξ 11139461

0wvtkdxle

α23(b + g)

2ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

minus (43)

+α23ϵξ2

bdminus 11 + ξ1113872 1113873 1113946

1

0uv

2tdx

+α235ϵ

ξ2 11139461

0u5dx + 21113946

1

0v5dx + 21113946

1

0w

5dx1113888 1113889 +α23gdminus 1

12ϵξ 1113946

1

0vv

2tdx

η11139461

0wthdxle

s

ϵd

minus 11 d

minus 12 M0 +

r

2ϵM

231 d

minus (43)2 ηξ 1113946

1

0w

5dx1113888 1113889

13

+e

2ϵM

231 d

minus (43)2 ηξ 1113946

1

0w

5dx1113888 1113889

13ηϵ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

α31ξ 11139461

0uwthdxle

α31s2ϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0u5dx1113888 1113889

13

+α31(r + 2e)

5ϵξ2 1113946

1

0u5dx1113888 1113889

13

+α31(3r + e)

10ϵξ2 1113946

1

0w

5dx1113888 1113889

13

+α31ξϵ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

14 Complexity

α32ξ 11139461

0vwthdxle

α32s2ϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0v5dx1113888 1113889

13

+2α32e5ϵ

ξ2 11139461

0u5dx +

α32(r + 2e)

5ϵξ2 1113946

1

0v5dx

+α32(e +(3r2))

5ϵξ2 1113946

1

0w

5dx +α32ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

2α33ξ 11139461

0wwthdxle

α33sϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0w

5dx1113888 1113889

13

+2α33e5ϵ

ξ2

+α33(5r + 3e)

5ϵξ2 1113946

1

0w

5dx + α33ϵξ sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

α12ξ 11139461

0uvtfdxle

α12ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+α125ϵ

2a +32

1113874 1113875ξ2 11139461

0u5dx

+aα1210ϵ

ξ2 11139461

0v5dx +

α125ϵ

ξ2 11139461

0w

5dx +α12ϵξ2

dminus 11 + ξ1113872 1113873 1113946

1

0uv

2tdx +

α12ϵξ2

dminus 11 + aξ1113872 1113873 1113946

1

0vv

2tdx

α12ξ 11139461

0uwtfdxle

α12ξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+ 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+3α12(a + 1)

10ϵξ2 1113946

1

0u5dx +

aα125ϵ

ξ2 11139461

0v5dx +

α125ϵ

ξ2 11139461

0w

5dx + α12ξϵ dminus 11 +

aξ2

+ξ2

1113888 1113889 11139461

0uw

2tdx

α21ξ 11139461

0vutkdxle

α21ξ2ϵ

M231 d

minus 11 d

minus (43)2 b 1113946

1

0u5dx1113888 1113889

13

+ g 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+α215ϵ

ξ2 11139461

0u5dx +

3α2110ϵ

ξ2 11139461

0v5dx +

α21ϵξ2

bdminus 11 + gd

minus 11 + ξ1113872 1113873 1113946

1

0vu

2tdx

α23ξ 11139461

0vwtkdxle

α23ξ2ϵ

M231 d

minus 11 d

minus (43)2 b 1113946

1

0u5dx1113888 1113889

13

+ g 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+α235ϵ

ξ2 11139461

0u5dx +

3α2310ϵ

ξ2 11139461

0v5dx +

α23ϵξ2

bdminus 11 + gd

minus 11 + ξ1113872 1113873 1113946

1

0vw

2tdx

α31ξ 11139461

0wuthdxle

α31sξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α31e5ϵ

ξ2 11139461

0u5dx

+α31ξ

2

2ϵr +

35

e1113874 1113875 11139461

0w

5dx +α31ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0wu

2tdx

α32ξ 11139461

0wvthdxle

α32sξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α32e5ϵ

ξ2 11139461

0u5dx

+α32ξ

2

2ϵr +

35

e1113874 1113875 11139461

0w

5dx +α32ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0wv

2tdx

(75)

Complexity 15

erefore

11139461

01 + p11ξ( 1113857utfdx + ξ 1113946

1

01 + p22( 1113857vtkdx + 1113946

1

0η + p33ξ( 1113857wthdx

+ ξ 11139461

0p12vtfdx + ξ 1113946

1

0p13wtfdx + ξ 1113946

1

0p21utkdx

+ ξ 11139461

0p23wtkdx + ξ 1113946

1

0p31uthdx + ξ 1113946

1

0p32vthdx

le λϵξ 11139461

0(u + v + w) u

2t + v

2t + w

2t1113872 1113873dx +

C8

ϵM0d

minus 11 d

minus 22 (1 + ξ + η)

+C9

ϵM

231 d

minus (43)2 ξ 1 + d

minus 11 + ξ + η1113872 1113873 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx1113890 1113891

13

+C10

ϵξ2 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx

(76)

where λ is a positive constant Choose a small enoughpositive number ε ε(di αij i j 1 2 3 a b g s

r e) such that λεltC4 Combining (74) and (76) onecan obtain12yprime(t)le minus

12

11139461

0P2xxdx minus

ξ2

11139461

0Q

2xxdx minus

η2

11139461

0R2xxdx

+ B1K2dminus 11 d

minus 12 + B2K3d

minus (43)1 z

13

+ B3K4dminus (23)1 z

23+ B4K5z

(77)

where z 111393810(u5 + v5 + w5)dx K2 (1 + ξ + η)

(M0 + M1dminus 11 dminus 1

2 ) K3 M231 ξ(1 + dminus 1

1 + ξ + η)

K4 M121 ξ2(1 + η) andK5 ξ2 Clearly Pge α11

ξu2 Qge α22ξv2 andRge α33ξw2 It follows from (49)that |P|5252 leC(|Px|2|P|321 + |P|521 ) and

zleB5ξminus (52)

11139461

0P52

+ Q52

+ R52

1113872 1113873dx

leB6ξminus (52)

K321 d

minus (32)1 y

12+ B6ξ

minus (52)K

521 d

minus (52)1

z13 leB7ξ

minus (56)K

121 d

minus (12)1 y

16+ B7ξ

minus (56)K

561 d

minus (56)1

z23 leB8ξ

minus (53)K1d

minus 11 y

13+ B8ξ

minus (53)K

531 d

minus (53)1

(78)

Moreover it follows by (50) that |Px|1032 leC(|Pxx|22|P|431 + |P|1031 )leB9K

431 d

minus (43)1 (|Pxx|22 + K2

1dminus 21 ) So

minusξ2

1113946

1

0

P2xxdx minus

12

1113946

1

0

Q2xxdx minus

η2

1113946

1

0

R2xxdx

le minus B10 min 1 ξ η1113864 1113865Kminus (43)1 d

431 y

53+(1 + ξ + η)K

21d

minus 21

(79)

Substituting the above inequality into (80) and thenmultiplying it by d2

2 we have

12

yprime(t)le minus A1 min 1 ξ η1113864 1113865Kminus (43)1 d

432 y

53

+ A2ξminus (56)

K121 K3d

minus (16)2 y

16

+ A3ξminus (53)

K1K4dminus (13)2 y

13

+ A4ξminus (52)

K321 K5d

minus (12)2 y

12

+ A5 K21(1 + ξ + η) + K2ξ1113960

+ K561 K3ξ

minus (56)d

minus (16)2 + K

531 K4ξ

minus (53)d

minus (12)2

+ K521 K5ξ

minus (52)d

minus (12)2 1113961

(80)

where y 111393810[(d2Px)2 + (d2Qx)2 + (d2Rx)2]dx

Iinequality (77) implies that there exist 1113957τ2 gt 0 and apositive constant 1113958M2 depending only ona b g s r e d1 d2 d3 αij(i j 1 2 3) such that

11139461

0d2Px( 1113857

2dx 11139461

0d2Qx( 1113857

2dx 11139461

0d2Rx( 1113857

2dxle 1113958M2 tge 1113957τ2

(81)

In the case that d1 d2 d3 ge 1 d2d1 d3d1 isin [d d]the coefficients of inequality (83) can be estimated bysome constants depending on d and d but doing noton d1 d2 andd3 So 1113958M2 depends on αij(i j

1 2 3) a b g r s e d d and is irrelevant to d1

d2 andd3 when d1 d2 d3 ge 1 d2d1 d3d1 isin [d d]Since

16 Complexity

ux

vx

wx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ J

minus 1

Px

Qx

Rx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

J

Pu Pv Pw

Qu Qv Qw

Ru Rv Rw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(82)

by detJgt ξη we have

d2ux

11138681113868111386811138681113868111386811138681113868 + d2vx

11138681113868111386811138681113868111386811138681113868 + d2wx

11138681113868111386811138681113868111386811138681113868leL d2Px

11138681113868111386811138681113868111386811138681113868 + d2Qx

11138681113868111386811138681113868111386811138681113868 + d2Rx

111386811138681113868111386811138681113868111386811138681113872 1113873

0lt xlt 1 tgt 0

(83)

where L is a constant depending only onξ η αij(i j 1 2 3) After scaling back and con-tacting estimates (84) and (85) there exist positiveconstants τ2 and M2 depending on αij(i j

1 2 3) di(i 1 2 3) a b g r s e such that

11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w

2xdxleM2 tge τ2 (84)

When d1 d2 d3 ge 1 d2d1 d3d1 isin [d d] M2 de-pends on d andd but do not on d1 d2 d3 ge 1

(2) tge 0 Modifying the dependency of the coefficients ininequalities (68)ndash(84) namely replacing M0 andM1with M0prime andM1prime there exists a positive constant M2primedepending on αij(i j 1 2 3) di(i 1 2 3) a b

g r s e and the W12minus norm of u0 v0 w0 such that

11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w

2xdxleM2prime tge 0 (85)

Furthermore when di ge 1(i 1 2 3) ξ η isin [ d d] M2primedepends on d andd but do not on di(i 1 2 3)

Summarizing estimates (54) (64) and (84) and theSobolev embedding theorem there exist positive constantsM andMprime depending only on αij(i j 1 2 3) di

(i 1 2 3) a b g r s e such that (51) and (54) hold Inparticular M andMprime depend only on αij(i j 1 2 3)

a b g r s e d d but do not on di(i 1 2 3) whendi ge 1(i 1 2 3) ξ η isin [ d d]

Similarly there exists a positive constant Mprimeprime dependingon αij(i j 1 2 3) di(i 1 2 3) a b g r s e and the ini-tial functions u0 v0 andw0 such that

|u(middot t)|12 |v(middot t)|12 |w(middot t)|12 leMprimeprime tge 0 (86)

Furthermore in the case that di ge 1(i 1 2 3)

ξ η isin [ d d] Mprimeprime depends only on d andd but do not ondi(i 1 2 3) us T +infin is completes the proof ofeorem 5

4 Asymptotic Behavior

In this section we will discuss the global asymptotic behaviorof the positive equilibrium point 1113957u for (6)

Theorem 6 Assume that all conditions in -eorem 5 aresatisfied Furthermore if aglt 1 and

4λρ1113957u1113957v1113957wd1d2d3 gt 1113957uM2 λα231113957v + ρα32 1113957w( 1113857

2d1 + 2α11M + α12M + α13M( 1113857

+ λ1113957vM2 α131113957u + ρα31 1113957w( 1113857

2d2 + α21M + 2α22M + α23M( 1113857

+ ρ1113957wM2 α121113957u + λα211113957v( 1113857

2d3 + α31M + α32M + 2α33M( 1113857

(87)

where λ ((2 minus ag)1113957u + g)g2 ρ 1e M is given by (58)then the positive equilibrium point 1113957u (1113957u 1113957v 1113957w) for (6) isglobally asymptotically stable

Proof Let (u v w) be a solution of (6) with u0 v0 w0 ge( equiv )0 From the strong maximum principle for parabolicequations we can prove that u v wgt 0 for any tgt 0

Define

V(u v w) 11139461

0u minus 1113957u minus 1113957u ln

u

1113957u1113874 1113875 + λ v minus 1113957v minus 1113957v ln

v

1113957v1113874 11138751113882

+ ρ w minus 1113957w minus 1113957w lnw

1113957w1113874 11138751113883dx

(88)

where λ ((2 minus ag)1113957u + g)g2 and ρ 1e ObviouslyV(u v w) is nonnegative and V(u v w) 0 if and only ifu 1113957u v 1113957v andw 1113957we time derivative ofV(u v w) forsystem (6) satisfies that

dV(u v w)

dt minus 1113946

1

0

1113957u

u2 d1 + p11( 1113857u2x +

λ1113957v

v2d2 + p22( 1113857v

2x1113896

+ρ1113957w

w2 d3 + p33( 1113857w2x +

α121113957u

u+λα211113957v

v1113888 1113889uxvx

+α131113957u

u+ρα31 1113957w

w1113874 1113875uxwx

+λα231113957v

v+ρα32 1113957w

w1113888 1113889vxwx1113897dx

minus 11139461

0

v

1113957uu(u minus 1113957u)

2+

bλu

1113957vv(v minus 1113957v)

21113896

+ ρr(w minus 1113957w)2

minusgλ + 1

1113957u+ a1113888 1113889(u minus 1113957u)(v minus 1113957v)

+(1 minus ρe)(u minus 1113957u)(w minus 1113957w)1113865dx

(89)

Complexity 17

It is easy to know that the second integrand in the aboveequality is positive definite by the choices of λ and ρ and thefirst integrand in the above equality is positive semidefinite if

4λρ1113957u1113957v1113957w d1 + p11( 1113857 d2 + p22( 1113857 d3 + p33( 1113857

+ α121113957uv + λα211113957vu( 1113857 α131113957uw + ρα31 1113957wu( 1113857 λα231113957vw + ρα32 1113957wv( 1113857

gt ρ1113957w α121113957uv + λα211113957vu( 11138572

d3 + p33( 1113857

+ λ1113957v α131113957uw + ρα31 1113957wu( 11138572

d2 + p22( 1113857

+ 1113957u λα231113957vw + ρα32 1113957wv( 11138572

d1 + p11( 1113857

(90)

By the maximum-norm estimate in eorem 5 condi-tion (87) implies that (90) holds erefore if the all con-ditions in eorem 6 hold there exists a positive constant δsuch thatdH(u v w)

dtle minus δ1113946

1

0(u minus 1113957u)

2+(v minus 1113957v)

2+(w minus 1113957w)

21113960 1113961dx

dH(u v w)

dtlt 0 (u v w)neE3

(91)

Using integration by parts the Holder inequality and(52) one can easily verify that ddt 1113938

10[(u minus 1113957u)2 + (v minus 1113957v)2 +

(w minus 1113957w)2]dx is bounded from above en from Lemma 1and (91) we have

|u(middot t) minus 1113957u|2⟶ 0

|v(middot t) minus 1113957v|2⟶ 0

|w(middot t) minus 1113957w|2⟶ 0

t⟶infin

(92)

It follows from the GagliardondashNirenberg inequality|u(middot t)|infin leC|u|1212 |u|122 that

|u(middot t) minus 1113957u|infin⟶ 0

|v(middot t) minus 1113957v|infin⟶ 0

|w(middot t) minus 1113957w|infin⟶ 0

t⟶infin

(93)

Namely (u v w) converges uniformly to (1113957u 1113957v 1113957w) Bythe fact that V(u v w) is decreasing for tge 0 it is obviousthat (1113957u 1113957v 1113957w) is globally asymptotically stable So the proofof eorem 6 is completed

Example 2 e following parameters satisfy all the con-ditions of eorem 6

α11 2

α12 1

α13 10

α21 32

α22 12

α23 1

α31 10

α32 2

α33 3

a 12

b 12

g 1

s 3

e 1

r 5

d1 d2 d3≫ 1

(94)

Remark 3 In model (6) if coefficients of reaction functionssatisfy the conditions of eorem 6 the diffusion matrix ispositive definite and the self-diffusion coefficients are largeenough then the model has no nonconstant positive steadystate

Data Availability

Data sharing is not applicable to this article as no datasetswere generated or analysed during the current study

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

18 Complexity

Acknowledgments

is research was supported by the National Natural ScienceFoundation of China (nos 11761063 and 11661051)

References

[1] K G Magnusson ldquoDestabilizing effect of cannibalism on astructured predator-prey systemrdquo Mathematical Biosciencesvol 155 no 1 pp 61ndash75 1999

[2] F van den Bosch A M de Roos and W Gabriel ldquoCanni-balism as a life boat mechanismrdquo Journal of MathematicalBiology vol 26 no 6 pp 619ndash633 1988

[3] M E Gurtin and D S Levine ldquoOn populations that can-nibalize their Youngrdquo SIAM Journal on Applied Mathematicsvol 42 no 1 pp 94ndash108 1982

[4] O Diekmann R M Nisbet W S C Gurney andF van den Bosch ldquoSimple mathematical models for canni-balism a critique and a new approachrdquo Mathematical Bio-sciences vol 78 no 1 pp 21ndash46 1986

[5] A Hastings ldquoCycles in cannibalistic egg-larval interactionsrdquoJournal of Mathematical Biology vol 24 no 6 pp 651ndash6661987

[6] C Kohlmeier and W Ebenhoh ldquoe stabilizing role ofcannibalism in a predator-prey systemrdquo Bulletin of Mathe-matical Biology vol 57 no 3 pp 401ndash411 1995

[7] F van den Bosch and W Gabriel ldquoCannibalism in an age-structured predator-prey systemrdquo Bulletin of MathematicalBiology vol 59 no 3 pp 551ndash567 1997

[8] J D Murray Mathematical Biology I An IntroductionSpringer New York NY USA 3rd edition 2003

[9] N Shigesada K Kawasaki and E Teramoto ldquoSpatial seg-regation of interacting speciesrdquo Journal of -eoretical Biologyvol 79 no 1 pp 83ndash99 1979

[10] S-A Shim ldquoUniform boundedness and convergence of so-lutions to the systems with cross-diffusions dominated by self-diffusionsrdquo Nonlinear Analysis Real World Applicationsvol 4 no 1 pp 65ndash86 2003

[11] Y S Choi R Lui and Y Yamada ldquoExistence of global so-lutions for the Shigesada-Kawasaki-Teramoto model withstrongly coupled cross-diffusionrdquo Discrete amp ContinuousDynamical Systems-A vol 10 no 3 pp 719ndash730 2004

[12] P Y H Pang and M Wang ldquoStrategy and stationary patternin a three-species predator-prey modelrdquo Journal of Differ-ential Equations vol 200 no 2 pp 245ndash273 2004

[13] S M Fu Z J Wen and S B Cui ldquoOn global solutions for thethree-species food-chain model with cross-diffusionrdquo ActaMathematica Sinica-A vol 50 pp 75ndash88 2007

[14] S M Fu and X L Yang ldquoNonconstant positive steady statesof a predator-prey model with cannibalismrdquo InternationalJournal of Information and Systems Sciences vol 8 pp 250ndash260 2012

[15] S Fu L Zhang and P Hu ldquoGlobal behavior of solutions in aLotka-Volterra predator-prey model with prey-stage struc-turerdquo Nonlinear Analysis Real World Applications vol 14no 5 pp 2027ndash2045 2013

[16] L Zhang and S Fu ldquoGlobal bifurcation for a Holling-Tannerpredator-prey model with prey-taxisrdquo Nonlinear AnalysisReal World Applications vol 47 pp 460ndash472 2019

[17] B Dubey B Das and J Hussain ldquoA predator-prey interactionmodel with self and cross-diffusionrdquo Ecological Modellingvol 141 no 1ndash3 pp 67ndash76 2001

[18] F Zhang Y Chen and J Li ldquoDynamical analysis of a stage-structured predator-prey model with cannibalismrdquo Mathe-matical Biosciences vol 307 pp 33ndash41 2019

[19] B Buonomo and D Lacitignola ldquoOn the stabilizing effect ofcannibalism in stage structured population modelsrdquo Math-ematical Biosciences and Engineering vol 3 pp 717ndash7312006

[20] K J Brown P C Dunne and R A Gardner ldquoA semilinearparabolic system arising in the theory of superconductivityrdquoJournal of Differential Equations vol 40 no 2 pp 232ndash2521981

[21] G Simonett ldquoCenter manifolds for quasilinear reaction-diffusion systemsrdquo Differential and Integral Equations vol 8pp 753ndash796 1995

[22] A-K Drangeid ldquoe principle of linearized stability forquasilinear parabolic evolution equationsrdquo Nonlinear Anal-ysis -eory Methods amp Applications vol 13 no 9pp 1091ndash1113 1989

[23] P Liu J Shi and Z-A Wang ldquoPattern formation of theattraction-repulsion Keller-Segel systemrdquo Discrete amp Con-tinuous Dynamical Systems-B vol 18 no 10 pp 2597ndash26252013

[24] H Amann ldquoDynamic theory of quasilinear parabolic equa-tions-I Abstract evolution equationsrdquo Nonlinear Analysis-eory Methods amp Applications vol 12 no 9 pp 895ndash9191988

[25] H Amann ldquoDynamic theory of quasilinear parabolic equa-tions-II Reaction-diffusionrdquo Differential and Integral Equa-tions vol 3 pp 13ndash75 1990

[26] H Amann ldquoDynamic theory of quasilinear parabolic sys-temsrdquoMathematische Zeitschrift vol 202 no 2 pp 219ndash2501989

Complexity 19

where

a BMmat

SA

cMmat

A

b RA

M2mat

g A + Mjuv

Mmat

s T

Mmat

r U

C

e V

S

(4)

We can see that e is the only parameter depending on S

in system (3)Most works on cannibalism adopt a McKendrickndashvon

Foerster age-structured model [2ndash4] but the model isusually rewritten as an ODE system or nonlinear Volterraintegral equations [4 5] In 1995 Kohlmeier and Ebenhoh[6] studied a two-dimensional ODE model without anystructure and they obtained that in some cases cannibalismcan lead to a higher long term predator stock size en vanden Bosch and Gabriel [7] found that cannibalism canstabilize a predator-prey system in a structured model wherethe oscillations are due to age structure

Magnusson [1] discussed model (1) which is in someways a simplification of the system studied by van den Boschand Gabriel in [7] Magnusson made two simplificationsFirstly it is assumed that all juveniles are vulnerable topredation by the adults Secondly instantaneous maturationinto the adult class is proportional to the present juvenilebiomass ie a constant per capita rate of maturation and

making this simplifying assumption means that any oscil-lations that may occur are not caused by a delay inherent inthe system Moreover the stability and Hopf bifurcation ofsolutions for system (3) were studied in [1] e authorobtained the following conclusions On one hand if themortality rate of juveniles is low andor the recruitment rateto the mature population is high then there is a stableequilibrium with all three population sizes as positive Onthe other hand if the mortality rate of juveniles is high andor the recruitment rate to the mature population is low thenthe equilibrium will be stable for low levels of cannibalismbut a loss of stability by a Hopf bifurcation will take place asthe level of cannibalism increases

As we all know the growth of biological populationdepends not only on time but also on spatial distributionSpatial species interaction includes the self-diffusion whichis the natural dispersive force of movement of individualand the cross-diffusion that is the population fluxes of onespecies due to the presence of other species [8ndash16] ere-fore taking into account the effect of competition for re-sources on species growth law we are naturally led to thefollowing weakly coupled reaction-diffusion model

ut d1Δu minus u + v + auv + uw x isin Ω tgt 0

vt d2Δv + bu minus gv minus uv x isin Ω tgt 0

wt d3Δw + sw minus rw2 minus euw x isin Ω tgt 0


u(x 0) u0(x)

v(x 0) v0(x)

w(x 0) w0(x)

x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)

and the Shigesada-Kawasaki-Teramoto model with stronglycoupled cross-diffusion

ut Δ d1u + α11u2 + α12uv + α13uw( 1113857 minus u + v + auv + uw x isin Ω tgt 0

vt Δ d2v + α21uv + α22v2 + α23vw( 1113857 + bu minus gv minus uv x isin Ω tgt 0

wt Δ d3w + α31uw + α32vw + α33w2( 1113857 + sw minus rw2 minus euw x isin Ω tgt 0


u(x 0) u0(x)

v(x 0) v0(x)

w(x 0) w0(x)

x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)

where Ω sub Rn is a bounded domain with smooth boundaryzΩ and η is the outward unit normal vector of the boundaryzΩ zη zzη e coefficients di(i 1 2 3) are the diffu-sion rates of u v andw αii(i 1 2 3) are referred as self-diffusion pressures and αij(ine j i j 1 2 3) are cross-diffusion coefficients which may be positive negative or

zero [17] e functions u0 v0 andw0 are nonnegativefunctions which are not identically zero

Very recently Zhang et al [18] found that the positiveequilibrium of (3) with r 0 can undergo stability switch(from stable to unstable to stable or from unstable to stableto unstable) with the change of the cannibalization rate

2 Complexity









u(x 0) u0(x)

v(x 0) v0(x)

w(x 0) w0(x)

x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)






u s

e

v bs

eg + s


eg + s

(8)




⎧⎪⎪⎨

⎪⎪⎩(9)


1113957u α +

α2 + 4eβ

1113968

2e

1113957v b1113957u

g + 1113957u

1113957w s minus e1113957u

r


β br + gs minus gr

(10)





Complexity 3

(4)







vs ws) reads

J us( 1113857

j11 j12 j13

j21 j22 j23

j31 j32 j33

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (11)


j12 1 + aus

j13 us

j21 b minus vs


j23 0

j31 minus ews

j32 0


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)



+ α2λ + α3 0 (13)

with

α1 11113957u

g1113957u + 1113957v + +1113957u2

+ r1113957u1113957w1113872 1113873gt 0

α2 11113957u


1113957w1113872 1113873

α3 11113957u


α1α2 minus α3 r1113957w1113957u2

+ (1 minus ag)1113957v + 2gr1113957w + re1113957w2

+ r2

1113957w2

1113872 11138731113957u

+ rg2

1113957w + e1113957v1113957w + 2r1113957v1113957w + gr2

1113957w2

+(1 minus ag)g1113957v +r1113957w1113957v2

1113957u2 +11113957u


+ r2

1113957w21113957v + 2gr1113957v1113957w1113872 1113873

(14)






X oplus+infin

i1Xi

Xi oplusdimE μi( )

j1Xij

(15)


Theorem 2













(16)



4 Complexity

I1 d21 d2 + d3( 1113857 + d

22 d1 + d3( 1113857 + d

23 d1 + d2( 1113857 + 2d1d2d3


231113872 1113873 minus j22 d

21 + d

231113872 1113873

minus 2 j11 + j22( 1113857 d1d2 + d2d3 + d1d3( 1113857

I3 j11 + j22( 1113857 d1j22 + d2j11 + d3j11 + d3j22( 1113857



(17)





+ Aiμ2i ξ2

+ Biμiξ +


limi⟶infin

1113957φi(ξ)

μ3i1113896 1113897 ξ3 + d1 + d2 + d3( 1113857ξ2

+ d1d2 + d2d3 + d1d3( 1113857ξ + d1d2d3 ≜ 1113957φ(ξ)

(19)





Theorem 3




A2 in [20] show




(20)


(1) Define

V(u v w) 1113946Ω



v

v1113874 11138751113876


w1113874 11138751113877dx

(21)


dV(u v w)

dt minus 1113946Ω

d1u

u2 |nablau|2

+λd2v

v2|nablav|

2+ρd3w

w2 |nablaw|2

1113888 1113889dx

minus1s

1113946Ω


s

v

u(u minus u)

2+


eg

u

v(v minus v)

2+

(e + as)

s

v

u

1113971

(u minus u) minus

s(e + as)u

v

1113970


2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭dx

le minus 1113946Ω

d1u

u2 |nablau|2

+λd2v

v2|nablav|

2+ρd3w

w2 |nablaw|2

1113888 1113889dx

minus1s

1113946Ω


s

v

u(u minus u)

2+


eg

u

v(v minus v)

21113896 1113897dx

(22)

Complexity 5


limt⟶infin

1113946Ω

|nablau|2

+|nablav|2

+|nablaw|2

1113872 1113873dx 0 (23)

limt⟶infin

1113946Ω

(u minus u)2dx 0

limt⟶infin

1113946Ω

(v minus v)2dx 0

(24)


limt⟶infin

1113946Ω

(u minus 1113954u)2dx 0

limt⟶infin

1113946Ω

(v minus 1113954v)2dx 0

limt⟶infin

1113946Ω

(w minus 1113954w)2dx 0

(25)



limt⟶infin

1113954u(t) u

limt⟶infin

1113954v(t) v(26)


1113954uprime tm( 1113857 1113946Ω



(27)


limtm⟶infin

1113954w(t) w (28)





m⟶infin

(29)



C2(Ω)⟶ 0


C2(Ω)⟶ 0


C2(Ω)⟶ 0

m⟶infin

(30)


(2) Define

V(u v w) 1113946Ω


u21113888 11138891113890


v21113888 1113889 +

1e

w1113891dx

(31)


dV(u v w)

dt minus 1113946Ω

d1u2

u2 |nablau|2

+λd2v2

v2|nablav|

21113888 1113889dx minus 1113946

Ω


v


21113896

+(1 minus ag)2 b


u


2+

r

ew

2

+1

b minus g

v

u

1113970


u

v

1113970

v minus v2( 11138571113888 1113889

2⎫⎬

⎭dx

(32)




6 Complexity



zuzt


zuz]

0 x isin zΩ tgt 0


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(33)


P

p11 p12 p13

p21 p22 p23

p31 p33 p33

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (34)

with

p11 2α11us + α12vs + α13ws

p12 α12us

p13 α13us

p21 α21vs

p22 α21us + 2α22vs + α23ws

p23 α23vs

p31 α31ws

p32 α32ws

p33 α31us + α32vs + 2α33ws

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(35)

We denote

L ≔ minus μi(D + P) + J (36)


ϕ λ3 + c1λ2

+ c2λ + c3 (37)

where




2i + β6μi + β7

(38)

with



β3 M22 + M33 + j22j33








(39)








Corollary 1



Complexity 7




wα23 + d2)


eg +


1113969







1113969







β1α13

21113957w3α23α33 + 21113957uα21α33 + 41113957vα22α33 + 2d3α33 + d3α23( 1113857 1113957w

2



β2α13




β3α13

(s minus 2e1113957u)(g + 1113957u)

(40)

Note that

limα13⟶infin

C α13 ]i( 1113857

α13 μi C1μ

2i + C2μi + C31113872 1113873 (41)


eg +


1113969


limα13⟶infin


C2 minus 4C1C3

1113968

2C1lt 0

limα13⟶infin

1113957μ2 0

limα13⟶infin

1113957μ3 minus C2 +

C2 minus 4C1C3

1113968

2C1gt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(42)





⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(43)







(1) Let

8 Complexity

α11 α12 α21 α22 α23 α32 1

α13 12

α31 3

α33 4

a 05

b 1

g 15

s 3

e 2

r 4

d1 3

d2 3

d3 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(44)


0326 0388) c1 5809gt 0 c2 10163gt 0 c3


(2) Let

α11 α12 α21 α22 α23 α31 α32 α33 1

d1 3

d2 3

d3 2

a 05

b 13

g 15

s 3

e 5

r 4

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(45)



(3) Let αij 01 i j 1 2 3 a 28 b 1 g 3 s










(46)






|u|2 leC ux

11138681113868111386811138681113868111386811138681113868132 |u|

231 + |u|11113874 1113875 forallu isinW

12(0 1) (47)

|u|4 leC ux

11138681113868111386811138681113868111386811138681113868122 |u|

121 + |u|11113874 1113875 forallu isinW

12(0 1) (48)

|u|52 leC ux

11138681113868111386811138681113868111386811138681113868252 |u|

351 + |u|11113874 1113875 forallu isinW

12(0 1) (49)

ux

111386811138681113868111386811138681113868111386811138682leC uxx

11138681113868111386811138681113868111386811138681113868352 |u|

251 + |u|11113874 1113875 forallu isinW

22(0 1) (50)



(H1) ablt 1 bltg




(51)


(52)


Complexity 9








ddt

11139461

0u + ρv +

1e

w1113874 1113875dxleC1 minus l 11139461

0u + ρv +

1e

w1113874 1113875dx

(53)


11139461

0udx 1113946

1

0vdx 1113946

1



11139461

0udx 1113946

1

0vdx 1113946

1



12ddt

11139461


1


1

0p11u

2xdx minus 1113946

1


1

0p13uxwxdx + 1113946

1

0uv + au

2v + u

2w1113872 1113873dx

12ddt

11139461


1


1

0p22v

2xdx minus 1113946

1


1

0p23vxwxdx + b 1113946

1

0uvdx

12ddt

11139461

0w


0w

2xdx minus 1113946

1

0p33w

2xdx minus 1113946

1


1

0p32vxwxdx + s 1113946

1

0w

2dx

(56)


12ddt

11139461

0u2

+ v2

+ w2


11139461

0u2x + v

2x + w

2x1113872 1113873dx minus 1113946

1


+(b + 1) 11139461

0uvdx + a 1113946

1

0u2vdx + s 1113946

1

0w

2dx

(57)

where

q ux vx wx( 1113857 p11u2x + p22v

2x + p33w

2x + p12 + p21( 1113857uxvx

+ p23 + p32( 1113857vxwx + p13 + p31( 1113857uxwx

(58)


12ddt

11139461

0u2

+ v2

+ w2


11139461

0u2x + v

2x + w

2x1113872 1113873dx

+(b + 1) 11139461

0uvdx

+ a 11139461

0u2vdx + s 1113946

1

0w

2dx

(59)

10 Complexity



|u|41)leCM20(|ux|22 + M2


a 11139461

0u2vdxle a 1113946

1

0

u4

2εdx + a 1113946

1

0

ε2v2dx

a2CM2

02dlowast

11139461

0v2dx +

dlowast

211139461

0u2xdx +

dlowastM20

2

(60)


11139461

0u2

+ v2

+ w2


211139461

0u2x + v

2x + w

2x1113872 1113873dx

+ C2 11139461

0u2

+ v2

+ w2

1113872 1113873dx +dlowastM2

02

(61)


|u|61 leCM40(|ux|22 + M2

0) erefore

minus 11139461

0u2x + v

2x + w

2x1113872 1113873dxle 3M

20 minus

19CM2

011139461

0u2

+ v2

+ w2

1113872 1113873dx1113890 1113891

3

(62)


12ddt

11139461

0u2

+ v2

+ w2


11139461

0u2

+ v2

+ w2

1113872 1113873dx1113890 1113891

3

+ C2 11139461

0u2

+ v2

+ w2

1113872 1113873dx + 2dlowastM

20

(63)


such that

11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w





11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w



1113957u u

d2

1113957v v

d2

1113957w w

d2

1113957t d1t

(66)






u(x 0) u0(x)

v(x 0) v0(x)

w(x 0) w0(x)

0ltxlt 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(67)




1u minus gdminus 1




11139461

0udx 1113946

1

0vdx 1113946

1

0wdxleM0d

minus 12

11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w

2dxleM1dminus 22


(68)


1


1

0v2tdx

minus η11139461

0w


1


+ 11139461


+ ξ11139461

0p21utk + 1+ p22( 1113857vtk + p23wtk1113858 1113859dx

+ 11139461


(69)

Complexity 11

where y 111393810(P2

x + Q2x + R2


1 2 3) such that


2t + w

2t1113872 1113873 (70)

erefore


1


1


1

0w


1

0(u + v + w) u

2t + v

2t + w

2t1113872 1113873dx

+ 11139461

01 + p11ξ( 1113857utfdx + 1113946

1

0ξ 1 + p22( 1113857vtkdx + 1113946

1

0η + p33ξ( 1113857wthdx

+ 11139461

0p12ξvtfdx + 1113946

1

0p13ξwtfdx + 1113946

1

0p21ξutkdx

+ 11139461

0p23ξwtkdx + 1113946

1

0p31ξuthdx + 1113946

1

0p32ξvthdx

(71)


11139461

0u4dxleM

131 d

minus (23)1 1113946

1

0u5dx1113888 1113889

23

11139461

0u2v2dxleM

131 d

minus (23)1 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

11139461

0u3dxleM

231 d

minus (43)1 1113946

1

0u5dx1113888 1113889

13

11139461

0uv

2dxleM231 d

minus (43)1 1113946

1

0v5dx1113888 1113889

13

11139461

0u3wdxleM

131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

12

11139461

0w

5dx1113888 1113889

16

11139461

0uvwdxleM

231 d

minus (43)2 1113946

1

0u5dx1113888 1113889

16

11139461

0v5dx1113888 1113889

16

11139461

0u2vwdxleM

131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

16

(72)



12 Complexity

minus 11139461

0u2tdxle minus

12

11139461

0P2xxdx + 1113946

1

0f2dx

minus ξ 11139461

0v2tdxle minus

ξ2

11139461

0Q

2xxdx + ξ 1113946

1

0k2dx

minus η11139461

0w

2tdxle minus

η2

11139461

0R2xxdx + η1113946

1

0h2dx

11139461

0f2dxle 2M1d

minus 21 d

minus 22 + a

2ξ2M131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

+ ξ2M131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0w

5dx1113888 1113889

13

+ 2aξdminus 11 d

minus (43)2 M

231 1113946

1

0v5dx1113888 1113889

13

+ 2ξdminus 11 d

minus (23)2 M

231 1113946

1

0u5dx1113888 1113889

16

11139461

0v5dx1113888 1113889

16


131 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

16

ξ 11139461

0k2dxle ξd

minus 21 d

minus 22 M1 b

2+ g

21113872 1113873 + ξ3dminus (23)

2 M131 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

+ ξ2gdminus 11 d

minus (43)2 M

231 1113946

1

0v5dx1113888 1113889

13

η11139461

0h2dxle ηs

2d

minus 21 d

minus 22 M1 + ηd

2ξ2dminus (23)2 M

131 1113946

1

0u5dx1113888 1113889

13

ηr2ξ2dminus (23)

2 M131


2 M131 1113946

1

0u5dx1113888 1113889

23


131 1113946

1

0u5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

12

(73)

us

minus 11139461


1


1

0w

2tdxle minus

12

11139461

0P2xxdx minus

ξ2

11139461

0Q

2xxdx minus

η2

11139461

0R2xxdx + C5(1 + ξ + η)d

minus 21 d

minus 22 M1

+ C6dminus (43)2 M

231 ξ(1 + η) 1113946

1

0u5

+ v5

1113872 1113873dx1113888 1113889

13

+ C7dminus (23)2 M

131 ξ2(1 + η) 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx1113888 1113889

23

(74)

Complexity 13

11139461

0utfdxle d

minus 11 1113946

1

0uutdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ d

minus 11 1113946

1

0utvdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ aξ 1113946

1

0uutvdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ ξ 1113946

1

0uutwdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le dminus 11

12ϵ

11139461

0udx +ϵ2

11139461

0uu

2tdx1113888 1113889 + d

minus 11

12ϵ

11139461

0vdx +ϵ2

11139461

0vu

2tdx1113888 1113889

+ aξ12ϵ

11139461

0uv

2dx +ϵ2

11139461

0uu

2tdx1113888 1113889 + ξ

12ϵ

11139461

0uw

2dx +ϵ2

11139461

0uu

2tdx1113888 1113889


2ϵ

M0 +a

2ϵM

231 d

minus (43)2 ξ 1113946

1

0v5dx1113888 1113889

13

+ξ2ϵ

M231 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+ϵdminus 1

12

11139461

0vu

2tdx +ϵ2

dminus 11 + aξ + ξ1113872 1113873 1113946

1

0uu

2tdx

2α11ξ 11139461

0uutfdxle

α11ϵ

M231 d

minus 11 d

232 1113946

1

0u5dx1113888 1113889

13

+ 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦ +

3α11(a + 1)

5ϵξ2 1113946

1

0u5dx

+2α11a5ϵ

ξ2 11139461

0v5dx +

2α115ϵ

ξ2 11139461

0w

5dx + α11ξϵ 2dminus 11 + aξ + ξ1113872 1113873 1113946

1

0uu

2tdx

2α12ξ 11139461

0vutfdxle

α12ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0v5dx1113888 1113889

13

+α12(a + 1)

5ϵξ2 1113946

1

0u5dx +

2α125ϵ

ξ2 11139461

0w

5dx +α12(3a + 4)

10ϵξ2 1113946

1

0v5dx

α13ξ 11139461

0wutfdxle

α13ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α13(a +(12))

5ϵξ2 1113946

1

0u5dx +

2aα135ϵ

ξ2 11139461

0v5dx

+2α13(a + 1)

5ϵξ2 1113946

1

0w

5dx +α13dminus 1

12ϵξ 1113946

1

0vu

2tdx +ϵ2


2+ 11113872 1113873 1113946

1

0uu

2tdx

ξ 11139461

0vtkdxle

b + g

2ϵM0d

minus 11 d

minus 12 ξ +

ξ2

2ϵM

231 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+bdminus 1

12ϵξ 1113946

1

0uv

2tdx +ϵ2ξ gd

minus 11 + ξ1113872 1113873 1113946

1

0vv

2tdx

α21ξ 11139461

0uvtkdxle

α21(b + g)

2ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+2α215ϵ

ξ2 11139461

0u5dx

+α2110ϵ

ξ2 11139461

0v5dx +

α21bdminus 11

2ϵξ 1113946

1

0uv

2tdx +

α21ϵξ2

gdminus 11 + ξ1113872 1113873 1113946

1

0vv

2tdx

2α22ξ 11139461

0vvtk dx le

α22(b + g)

ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+α225ϵ

ξ2 11139461

0u5dx

+4α225ϵ

ξ2 11139461

0v5dx + α22(b + g)d

minus 11 ϵξ 1113946

1

0vv

2tdx + α22ϵξ

211139461

0uv

2tdx

α23ξ 11139461

0wvtkdxle

α23(b + g)

2ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

minus (43)

+α23ϵξ2

bdminus 11 + ξ1113872 1113873 1113946

1

0uv

2tdx

+α235ϵ

ξ2 11139461

0u5dx + 21113946

1

0v5dx + 21113946

1

0w

5dx1113888 1113889 +α23gdminus 1

12ϵξ 1113946

1

0vv

2tdx

η11139461

0wthdxle

s

ϵd

minus 11 d

minus 12 M0 +

r

2ϵM

231 d

minus (43)2 ηξ 1113946

1

0w

5dx1113888 1113889

13

+e

2ϵM

231 d

minus (43)2 ηξ 1113946

1

0w

5dx1113888 1113889

13ηϵ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

α31ξ 11139461

0uwthdxle

α31s2ϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0u5dx1113888 1113889

13

+α31(r + 2e)

5ϵξ2 1113946

1

0u5dx1113888 1113889

13

+α31(3r + e)

10ϵξ2 1113946

1

0w

5dx1113888 1113889

13

+α31ξϵ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

14 Complexity

α32ξ 11139461

0vwthdxle

α32s2ϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0v5dx1113888 1113889

13

+2α32e5ϵ

ξ2 11139461

0u5dx +

α32(r + 2e)

5ϵξ2 1113946

1

0v5dx

+α32(e +(3r2))

5ϵξ2 1113946

1

0w

5dx +α32ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

2α33ξ 11139461

0wwthdxle

α33sϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0w

5dx1113888 1113889

13

+2α33e5ϵ

ξ2

+α33(5r + 3e)

5ϵξ2 1113946

1

0w


1

0ww

2tdx

α12ξ 11139461

0uvtfdxle

α12ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+α125ϵ

2a +32

1113874 1113875ξ2 11139461

0u5dx

+aα1210ϵ

ξ2 11139461

0v5dx +

α125ϵ

ξ2 11139461

0w

5dx +α12ϵξ2

dminus 11 + ξ1113872 1113873 1113946

1

0uv

2tdx +

α12ϵξ2

dminus 11 + aξ1113872 1113873 1113946

1

0vv

2tdx

α12ξ 11139461

0uwtfdxle

α12ξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+ 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+3α12(a + 1)

10ϵξ2 1113946

1

0u5dx +

aα125ϵ

ξ2 11139461

0v5dx +

α125ϵ

ξ2 11139461

0w


aξ2

+ξ2

1113888 1113889 11139461

0uw

2tdx

α21ξ 11139461

0vutkdxle

α21ξ2ϵ

M231 d

minus 11 d

minus (43)2 b 1113946

1

0u5dx1113888 1113889

13

+ g 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+α215ϵ

ξ2 11139461

0u5dx +

3α2110ϵ

ξ2 11139461

0v5dx +

α21ϵξ2

bdminus 11 + gd

minus 11 + ξ1113872 1113873 1113946

1

0vu

2tdx

α23ξ 11139461

0vwtkdxle

α23ξ2ϵ

M231 d

minus 11 d

minus (43)2 b 1113946

1

0u5dx1113888 1113889

13

+ g 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+α235ϵ

ξ2 11139461

0u5dx +

3α2310ϵ

ξ2 11139461

0v5dx +

α23ϵξ2

bdminus 11 + gd

minus 11 + ξ1113872 1113873 1113946

1

0vw

2tdx

α31ξ 11139461

0wuthdxle

α31sξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α31e5ϵ

ξ2 11139461

0u5dx

+α31ξ

2

2ϵr +

35

e1113874 1113875 11139461

0w

5dx +α31ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0wu

2tdx

α32ξ 11139461

0wvthdxle

α32sξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α32e5ϵ

ξ2 11139461

0u5dx

+α32ξ

2

2ϵr +

35

e1113874 1113875 11139461

0w

5dx +α32ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0wv

2tdx

(75)

Complexity 15

erefore

11139461

01 + p11ξ( 1113857utfdx + ξ 1113946

1

01 + p22( 1113857vtkdx + 1113946

1

0η + p33ξ( 1113857wthdx

+ ξ 11139461

0p12vtfdx + ξ 1113946

1

0p13wtfdx + ξ 1113946

1

0p21utkdx

+ ξ 11139461

0p23wtkdx + ξ 1113946

1

0p31uthdx + ξ 1113946

1

0p32vthdx

le λϵξ 11139461

0(u + v + w) u

2t + v

2t + w

2t1113872 1113873dx +

C8

ϵM0d

minus 11 d

minus 22 (1 + ξ + η)

+C9

ϵM

231 d

minus (43)2 ξ 1 + d

minus 11 + ξ + η1113872 1113873 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx1113890 1113891

13

+C10

ϵξ2 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx

(76)



12

11139461

0P2xxdx minus

ξ2

11139461

0Q

2xxdx minus

η2

11139461

0R2xxdx

+ B1K2dminus 11 d

minus 12 + B2K3d

minus (43)1 z

13


23+ B4K5z

(77)

where z 111393810(u5 + v5 + w5)dx K2 (1 + ξ + η)


2 ) K3 M231 ξ(1 + dminus 1

1 + ξ + η)



zleB5ξminus (52)

11139461

0P52

+ Q52

+ R52

1113872 1113873dx

leB6ξminus (52)

K321 d

minus (32)1 y

12+ B6ξ

minus (52)K

521 d

minus (52)1

z13 leB7ξ

minus (56)K

121 d

minus (12)1 y

16+ B7ξ

minus (56)K

561 d

minus (56)1

z23 leB8ξ

minus (53)K1d

minus 11 y

13+ B8ξ

minus (53)K

531 d

minus (53)1

(78)


431 d

minus (43)1 (|Pxx|22 + K2

1dminus 21 ) So

minusξ2

1113946

1

0

P2xxdx minus

12

1113946

1

0

Q2xxdx minus

η2

1113946

1

0

R2xxdx


431 y

53+(1 + ξ + η)K

21d

minus 21

(79)


2 we have

12


432 y

53

+ A2ξminus (56)

K121 K3d

minus (16)2 y

16

+ A3ξminus (53)

K1K4dminus (13)2 y

13

+ A4ξminus (52)

K321 K5d

minus (12)2 y

12

+ A5 K21(1 + ξ + η) + K2ξ1113960

+ K561 K3ξ

minus (56)d

minus (16)2 + K

531 K4ξ

minus (53)d

minus (12)2

+ K521 K5ξ

minus (52)d

minus (12)2 1113961

(80)



11139461

0d2Px( 1113857

2dx 11139461

0d2Qx( 1113857

2dx 11139461

0d2Rx( 1113857

2dxle 1113958M2 tge 1113957τ2

(81)




16 Complexity

ux

vx

wx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ J

minus 1

Px

Qx

Rx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

J

Pu Pv Pw

Qu Qv Qw

Ru Rv Rw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(82)


d2ux

11138681113868111386811138681113868111386811138681113868 + d2vx

11138681113868111386811138681113868111386811138681113868 + d2wx

11138681113868111386811138681113868111386811138681113868leL d2Px

11138681113868111386811138681113868111386811138681113868 + d2Qx

11138681113868111386811138681113868111386811138681113868 + d2Rx

111386811138681113868111386811138681113868111386811138681113872 1113873

0lt xlt 1 tgt 0

(83)



11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w





11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w














2d1 + 2α11M + α12M + α13M( 1113857

+ λ1113957vM2 α131113957u + ρα31 1113957w( 1113857

2d2 + α21M + 2α22M + α23M( 1113857

+ ρ1113957wM2 α121113957u + λα211113957v( 1113857

2d3 + α31M + α32M + 2α33M( 1113857

(87)



Define

V(u v w) 11139461


u


v

1113957v1113874 11138751113882


1113957w1113874 11138751113883dx

(88)


dV(u v w)

dt minus 1113946

1

0

1113957u

u2 d1 + p11( 1113857u2x +

λ1113957v

v2d2 + p22( 1113857v

2x1113896

+ρ1113957w

w2 d3 + p33( 1113857w2x +

α121113957u

u+λα211113957v

v1113888 1113889uxvx

+α131113957u

u+ρα31 1113957w

w1113874 1113875uxwx

+λα231113957v

v+ρα32 1113957w

w1113888 1113889vxwx1113897dx

minus 11139461

0

v

1113957uu(u minus 1113957u)

2+

bλu

1113957vv(v minus 1113957v)

21113896

+ ρr(w minus 1113957w)2

minusgλ + 1



(89)

Complexity 17


4λρ1113957u1113957v1113957w d1 + p11( 1113857 d2 + p22( 1113857 d3 + p33( 1113857


gt ρ1113957w α121113957uv + λα211113957vu( 11138572

d3 + p33( 1113857

+ λ1113957v α131113957uw + ρα31 1113957wu( 11138572

d2 + p22( 1113857

+ 1113957u λα231113957vw + ρα32 1113957wv( 11138572

d1 + p11( 1113857

(90)



1

0(u minus 1113957u)

2+(v minus 1113957v)

2+(w minus 1113957w)

21113960 1113961dx

dH(u v w)

dtlt 0 (u v w)neE3

(91)


10[(u minus 1113957u)2 + (v minus 1113957v)2 +





t⟶infin

(92)





t⟶infin

(93)



α11 2

α12 1

α13 10

α21 32

α22 12

α23 1

α31 10

α32 2

α33 3

a 12

b 12

g 1

s 3

e 1

r 5

d1 d2 d3≫ 1

(94)


Data Availability




18 Complexity

Acknowledgments


References



























Complexity 19









u(x 0) u0(x)

v(x 0) v0(x)

w(x 0) w0(x)

x isin Ω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)






u s

e

v bs

eg + s


eg + s

(8)




⎧⎪⎪⎨

⎪⎪⎩(9)


1113957u α +

α2 + 4eβ

1113968

2e

1113957v b1113957u

g + 1113957u

1113957w s minus e1113957u

r


β br + gs minus gr

(10)





Complexity 3

(4)







vs ws) reads

J us( 1113857

j11 j12 j13

j21 j22 j23

j31 j32 j33

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (11)


j12 1 + aus

j13 us

j21 b minus vs


j23 0

j31 minus ews

j32 0


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)



+ α2λ + α3 0 (13)

with

α1 11113957u

g1113957u + 1113957v + +1113957u2

+ r1113957u1113957w1113872 1113873gt 0

α2 11113957u


1113957w1113872 1113873

α3 11113957u


α1α2 minus α3 r1113957w1113957u2

+ (1 minus ag)1113957v + 2gr1113957w + re1113957w2

+ r2

1113957w2

1113872 11138731113957u

+ rg2

1113957w + e1113957v1113957w + 2r1113957v1113957w + gr2

1113957w2

+(1 minus ag)g1113957v +r1113957w1113957v2

1113957u2 +11113957u


+ r2

1113957w21113957v + 2gr1113957v1113957w1113872 1113873

(14)






X oplus+infin

i1Xi

Xi oplusdimE μi( )

j1Xij

(15)


Theorem 2













(16)



4 Complexity

I1 d21 d2 + d3( 1113857 + d

22 d1 + d3( 1113857 + d

23 d1 + d2( 1113857 + 2d1d2d3


231113872 1113873 minus j22 d

21 + d

231113872 1113873

minus 2 j11 + j22( 1113857 d1d2 + d2d3 + d1d3( 1113857

I3 j11 + j22( 1113857 d1j22 + d2j11 + d3j11 + d3j22( 1113857



(17)





+ Aiμ2i ξ2

+ Biμiξ +


limi⟶infin

1113957φi(ξ)

μ3i1113896 1113897 ξ3 + d1 + d2 + d3( 1113857ξ2

+ d1d2 + d2d3 + d1d3( 1113857ξ + d1d2d3 ≜ 1113957φ(ξ)

(19)





Theorem 3




A2 in [20] show




(20)


(1) Define

V(u v w) 1113946Ω



v

v1113874 11138751113876


w1113874 11138751113877dx

(21)


dV(u v w)

dt minus 1113946Ω

d1u

u2 |nablau|2

+λd2v

v2|nablav|

2+ρd3w

w2 |nablaw|2

1113888 1113889dx

minus1s

1113946Ω


s

v

u(u minus u)

2+


eg

u

v(v minus v)

2+

(e + as)

s

v

u

1113971

(u minus u) minus

s(e + as)u

v

1113970


2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭dx

le minus 1113946Ω

d1u

u2 |nablau|2

+λd2v

v2|nablav|

2+ρd3w

w2 |nablaw|2

1113888 1113889dx

minus1s

1113946Ω


s

v

u(u minus u)

2+


eg

u

v(v minus v)

21113896 1113897dx

(22)

Complexity 5


limt⟶infin

1113946Ω

|nablau|2

+|nablav|2

+|nablaw|2

1113872 1113873dx 0 (23)

limt⟶infin

1113946Ω

(u minus u)2dx 0

limt⟶infin

1113946Ω

(v minus v)2dx 0

(24)


limt⟶infin

1113946Ω

(u minus 1113954u)2dx 0

limt⟶infin

1113946Ω

(v minus 1113954v)2dx 0

limt⟶infin

1113946Ω

(w minus 1113954w)2dx 0

(25)



limt⟶infin

1113954u(t) u

limt⟶infin

1113954v(t) v(26)


1113954uprime tm( 1113857 1113946Ω



(27)


limtm⟶infin

1113954w(t) w (28)





m⟶infin

(29)



C2(Ω)⟶ 0


C2(Ω)⟶ 0


C2(Ω)⟶ 0

m⟶infin

(30)


(2) Define

V(u v w) 1113946Ω


u21113888 11138891113890


v21113888 1113889 +

1e

w1113891dx

(31)


dV(u v w)

dt minus 1113946Ω

d1u2

u2 |nablau|2

+λd2v2

v2|nablav|

21113888 1113889dx minus 1113946

Ω


v


21113896

+(1 minus ag)2 b


u


2+

r

ew

2

+1

b minus g

v

u

1113970


u

v

1113970

v minus v2( 11138571113888 1113889

2⎫⎬

⎭dx

(32)




6 Complexity



zuzt


zuz]

0 x isin zΩ tgt 0


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(33)


P

p11 p12 p13

p21 p22 p23

p31 p33 p33

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (34)

with

p11 2α11us + α12vs + α13ws

p12 α12us

p13 α13us

p21 α21vs

p22 α21us + 2α22vs + α23ws

p23 α23vs

p31 α31ws

p32 α32ws

p33 α31us + α32vs + 2α33ws

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(35)

We denote

L ≔ minus μi(D + P) + J (36)


ϕ λ3 + c1λ2

+ c2λ + c3 (37)

where




2i + β6μi + β7

(38)

with



β3 M22 + M33 + j22j33








(39)








Corollary 1



Complexity 7




wα23 + d2)


eg +


1113969







1113969







β1α13

21113957w3α23α33 + 21113957uα21α33 + 41113957vα22α33 + 2d3α33 + d3α23( 1113857 1113957w

2



β2α13




β3α13

(s minus 2e1113957u)(g + 1113957u)

(40)

Note that

limα13⟶infin

C α13 ]i( 1113857

α13 μi C1μ

2i + C2μi + C31113872 1113873 (41)


eg +


1113969


limα13⟶infin


C2 minus 4C1C3

1113968

2C1lt 0

limα13⟶infin

1113957μ2 0

limα13⟶infin

1113957μ3 minus C2 +

C2 minus 4C1C3

1113968

2C1gt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(42)





⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(43)







(1) Let

8 Complexity

α11 α12 α21 α22 α23 α32 1

α13 12

α31 3

α33 4

a 05

b 1

g 15

s 3

e 2

r 4

d1 3

d2 3

d3 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(44)


0326 0388) c1 5809gt 0 c2 10163gt 0 c3


(2) Let

α11 α12 α21 α22 α23 α31 α32 α33 1

d1 3

d2 3

d3 2

a 05

b 13

g 15

s 3

e 5

r 4

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(45)



(3) Let αij 01 i j 1 2 3 a 28 b 1 g 3 s










(46)






|u|2 leC ux

11138681113868111386811138681113868111386811138681113868132 |u|

231 + |u|11113874 1113875 forallu isinW

12(0 1) (47)

|u|4 leC ux

11138681113868111386811138681113868111386811138681113868122 |u|

121 + |u|11113874 1113875 forallu isinW

12(0 1) (48)

|u|52 leC ux

11138681113868111386811138681113868111386811138681113868252 |u|

351 + |u|11113874 1113875 forallu isinW

12(0 1) (49)

ux

111386811138681113868111386811138681113868111386811138682leC uxx

11138681113868111386811138681113868111386811138681113868352 |u|

251 + |u|11113874 1113875 forallu isinW

22(0 1) (50)



(H1) ablt 1 bltg




(51)


(52)


Complexity 9








ddt

11139461

0u + ρv +

1e

w1113874 1113875dxleC1 minus l 11139461

0u + ρv +

1e

w1113874 1113875dx

(53)


11139461

0udx 1113946

1

0vdx 1113946

1



11139461

0udx 1113946

1

0vdx 1113946

1



12ddt

11139461


1


1

0p11u

2xdx minus 1113946

1


1

0p13uxwxdx + 1113946

1

0uv + au

2v + u

2w1113872 1113873dx

12ddt

11139461


1


1

0p22v

2xdx minus 1113946

1


1

0p23vxwxdx + b 1113946

1

0uvdx

12ddt

11139461

0w


0w

2xdx minus 1113946

1

0p33w

2xdx minus 1113946

1


1

0p32vxwxdx + s 1113946

1

0w

2dx

(56)


12ddt

11139461

0u2

+ v2

+ w2


11139461

0u2x + v

2x + w

2x1113872 1113873dx minus 1113946

1


+(b + 1) 11139461

0uvdx + a 1113946

1

0u2vdx + s 1113946

1

0w

2dx

(57)

where

q ux vx wx( 1113857 p11u2x + p22v

2x + p33w

2x + p12 + p21( 1113857uxvx

+ p23 + p32( 1113857vxwx + p13 + p31( 1113857uxwx

(58)


12ddt

11139461

0u2

+ v2

+ w2


11139461

0u2x + v

2x + w

2x1113872 1113873dx

+(b + 1) 11139461

0uvdx

+ a 11139461

0u2vdx + s 1113946

1

0w

2dx

(59)

10 Complexity



|u|41)leCM20(|ux|22 + M2


a 11139461

0u2vdxle a 1113946

1

0

u4

2εdx + a 1113946

1

0

ε2v2dx

a2CM2

02dlowast

11139461

0v2dx +

dlowast

211139461

0u2xdx +

dlowastM20

2

(60)


11139461

0u2

+ v2

+ w2


211139461

0u2x + v

2x + w

2x1113872 1113873dx

+ C2 11139461

0u2

+ v2

+ w2

1113872 1113873dx +dlowastM2

02

(61)


|u|61 leCM40(|ux|22 + M2

0) erefore

minus 11139461

0u2x + v

2x + w

2x1113872 1113873dxle 3M

20 minus

19CM2

011139461

0u2

+ v2

+ w2

1113872 1113873dx1113890 1113891

3

(62)


12ddt

11139461

0u2

+ v2

+ w2


11139461

0u2

+ v2

+ w2

1113872 1113873dx1113890 1113891

3

+ C2 11139461

0u2

+ v2

+ w2

1113872 1113873dx + 2dlowastM

20

(63)


such that

11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w





11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w



1113957u u

d2

1113957v v

d2

1113957w w

d2

1113957t d1t

(66)






u(x 0) u0(x)

v(x 0) v0(x)

w(x 0) w0(x)

0ltxlt 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(67)




1u minus gdminus 1




11139461

0udx 1113946

1

0vdx 1113946

1

0wdxleM0d

minus 12

11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w

2dxleM1dminus 22


(68)


1


1

0v2tdx

minus η11139461

0w


1


+ 11139461


+ ξ11139461

0p21utk + 1+ p22( 1113857vtk + p23wtk1113858 1113859dx

+ 11139461


(69)

Complexity 11

where y 111393810(P2

x + Q2x + R2


1 2 3) such that


2t + w

2t1113872 1113873 (70)

erefore


1


1


1

0w


1

0(u + v + w) u

2t + v

2t + w

2t1113872 1113873dx

+ 11139461

01 + p11ξ( 1113857utfdx + 1113946

1

0ξ 1 + p22( 1113857vtkdx + 1113946

1

0η + p33ξ( 1113857wthdx

+ 11139461

0p12ξvtfdx + 1113946

1

0p13ξwtfdx + 1113946

1

0p21ξutkdx

+ 11139461

0p23ξwtkdx + 1113946

1

0p31ξuthdx + 1113946

1

0p32ξvthdx

(71)


11139461

0u4dxleM

131 d

minus (23)1 1113946

1

0u5dx1113888 1113889

23

11139461

0u2v2dxleM

131 d

minus (23)1 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

11139461

0u3dxleM

231 d

minus (43)1 1113946

1

0u5dx1113888 1113889

13

11139461

0uv

2dxleM231 d

minus (43)1 1113946

1

0v5dx1113888 1113889

13

11139461

0u3wdxleM

131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

12

11139461

0w

5dx1113888 1113889

16

11139461

0uvwdxleM

231 d

minus (43)2 1113946

1

0u5dx1113888 1113889

16

11139461

0v5dx1113888 1113889

16

11139461

0u2vwdxleM

131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

16

(72)



12 Complexity

minus 11139461

0u2tdxle minus

12

11139461

0P2xxdx + 1113946

1

0f2dx

minus ξ 11139461

0v2tdxle minus

ξ2

11139461

0Q

2xxdx + ξ 1113946

1

0k2dx

minus η11139461

0w

2tdxle minus

η2

11139461

0R2xxdx + η1113946

1

0h2dx

11139461

0f2dxle 2M1d

minus 21 d

minus 22 + a

2ξ2M131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

+ ξ2M131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0w

5dx1113888 1113889

13

+ 2aξdminus 11 d

minus (43)2 M

231 1113946

1

0v5dx1113888 1113889

13

+ 2ξdminus 11 d

minus (23)2 M

231 1113946

1

0u5dx1113888 1113889

16

11139461

0v5dx1113888 1113889

16


131 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

16

ξ 11139461

0k2dxle ξd

minus 21 d

minus 22 M1 b

2+ g

21113872 1113873 + ξ3dminus (23)

2 M131 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

+ ξ2gdminus 11 d

minus (43)2 M

231 1113946

1

0v5dx1113888 1113889

13

η11139461

0h2dxle ηs

2d

minus 21 d

minus 22 M1 + ηd

2ξ2dminus (23)2 M

131 1113946

1

0u5dx1113888 1113889

13

ηr2ξ2dminus (23)

2 M131


2 M131 1113946

1

0u5dx1113888 1113889

23


131 1113946

1

0u5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

12

(73)

us

minus 11139461


1


1

0w

2tdxle minus

12

11139461

0P2xxdx minus

ξ2

11139461

0Q

2xxdx minus

η2

11139461

0R2xxdx + C5(1 + ξ + η)d

minus 21 d

minus 22 M1

+ C6dminus (43)2 M

231 ξ(1 + η) 1113946

1

0u5

+ v5

1113872 1113873dx1113888 1113889

13

+ C7dminus (23)2 M

131 ξ2(1 + η) 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx1113888 1113889

23

(74)

Complexity 13

11139461

0utfdxle d

minus 11 1113946

1

0uutdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ d

minus 11 1113946

1

0utvdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ aξ 1113946

1

0uutvdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ ξ 1113946

1

0uutwdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le dminus 11

12ϵ

11139461

0udx +ϵ2

11139461

0uu

2tdx1113888 1113889 + d

minus 11

12ϵ

11139461

0vdx +ϵ2

11139461

0vu

2tdx1113888 1113889

+ aξ12ϵ

11139461

0uv

2dx +ϵ2

11139461

0uu

2tdx1113888 1113889 + ξ

12ϵ

11139461

0uw

2dx +ϵ2

11139461

0uu

2tdx1113888 1113889


2ϵ

M0 +a

2ϵM

231 d

minus (43)2 ξ 1113946

1

0v5dx1113888 1113889

13

+ξ2ϵ

M231 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+ϵdminus 1

12

11139461

0vu

2tdx +ϵ2

dminus 11 + aξ + ξ1113872 1113873 1113946

1

0uu

2tdx

2α11ξ 11139461

0uutfdxle

α11ϵ

M231 d

minus 11 d

232 1113946

1

0u5dx1113888 1113889

13

+ 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦ +

3α11(a + 1)

5ϵξ2 1113946

1

0u5dx

+2α11a5ϵ

ξ2 11139461

0v5dx +

2α115ϵ

ξ2 11139461

0w

5dx + α11ξϵ 2dminus 11 + aξ + ξ1113872 1113873 1113946

1

0uu

2tdx

2α12ξ 11139461

0vutfdxle

α12ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0v5dx1113888 1113889

13

+α12(a + 1)

5ϵξ2 1113946

1

0u5dx +

2α125ϵ

ξ2 11139461

0w

5dx +α12(3a + 4)

10ϵξ2 1113946

1

0v5dx

α13ξ 11139461

0wutfdxle

α13ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α13(a +(12))

5ϵξ2 1113946

1

0u5dx +

2aα135ϵ

ξ2 11139461

0v5dx

+2α13(a + 1)

5ϵξ2 1113946

1

0w

5dx +α13dminus 1

12ϵξ 1113946

1

0vu

2tdx +ϵ2


2+ 11113872 1113873 1113946

1

0uu

2tdx

ξ 11139461

0vtkdxle

b + g

2ϵM0d

minus 11 d

minus 12 ξ +

ξ2

2ϵM

231 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+bdminus 1

12ϵξ 1113946

1

0uv

2tdx +ϵ2ξ gd

minus 11 + ξ1113872 1113873 1113946

1

0vv

2tdx

α21ξ 11139461

0uvtkdxle

α21(b + g)

2ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+2α215ϵ

ξ2 11139461

0u5dx

+α2110ϵ

ξ2 11139461

0v5dx +

α21bdminus 11

2ϵξ 1113946

1

0uv

2tdx +

α21ϵξ2

gdminus 11 + ξ1113872 1113873 1113946

1

0vv

2tdx

2α22ξ 11139461

0vvtk dx le

α22(b + g)

ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+α225ϵ

ξ2 11139461

0u5dx

+4α225ϵ

ξ2 11139461

0v5dx + α22(b + g)d

minus 11 ϵξ 1113946

1

0vv

2tdx + α22ϵξ

211139461

0uv

2tdx

α23ξ 11139461

0wvtkdxle

α23(b + g)

2ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

minus (43)

+α23ϵξ2

bdminus 11 + ξ1113872 1113873 1113946

1

0uv

2tdx

+α235ϵ

ξ2 11139461

0u5dx + 21113946

1

0v5dx + 21113946

1

0w

5dx1113888 1113889 +α23gdminus 1

12ϵξ 1113946

1

0vv

2tdx

η11139461

0wthdxle

s

ϵd

minus 11 d

minus 12 M0 +

r

2ϵM

231 d

minus (43)2 ηξ 1113946

1

0w

5dx1113888 1113889

13

+e

2ϵM

231 d

minus (43)2 ηξ 1113946

1

0w

5dx1113888 1113889

13ηϵ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

α31ξ 11139461

0uwthdxle

α31s2ϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0u5dx1113888 1113889

13

+α31(r + 2e)

5ϵξ2 1113946

1

0u5dx1113888 1113889

13

+α31(3r + e)

10ϵξ2 1113946

1

0w

5dx1113888 1113889

13

+α31ξϵ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

14 Complexity

α32ξ 11139461

0vwthdxle

α32s2ϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0v5dx1113888 1113889

13

+2α32e5ϵ

ξ2 11139461

0u5dx +

α32(r + 2e)

5ϵξ2 1113946

1

0v5dx

+α32(e +(3r2))

5ϵξ2 1113946

1

0w

5dx +α32ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

2α33ξ 11139461

0wwthdxle

α33sϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0w

5dx1113888 1113889

13

+2α33e5ϵ

ξ2

+α33(5r + 3e)

5ϵξ2 1113946

1

0w


1

0ww

2tdx

α12ξ 11139461

0uvtfdxle

α12ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+α125ϵ

2a +32

1113874 1113875ξ2 11139461

0u5dx

+aα1210ϵ

ξ2 11139461

0v5dx +

α125ϵ

ξ2 11139461

0w

5dx +α12ϵξ2

dminus 11 + ξ1113872 1113873 1113946

1

0uv

2tdx +

α12ϵξ2

dminus 11 + aξ1113872 1113873 1113946

1

0vv

2tdx

α12ξ 11139461

0uwtfdxle

α12ξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+ 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+3α12(a + 1)

10ϵξ2 1113946

1

0u5dx +

aα125ϵ

ξ2 11139461

0v5dx +

α125ϵ

ξ2 11139461

0w


aξ2

+ξ2

1113888 1113889 11139461

0uw

2tdx

α21ξ 11139461

0vutkdxle

α21ξ2ϵ

M231 d

minus 11 d

minus (43)2 b 1113946

1

0u5dx1113888 1113889

13

+ g 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+α215ϵ

ξ2 11139461

0u5dx +

3α2110ϵ

ξ2 11139461

0v5dx +

α21ϵξ2

bdminus 11 + gd

minus 11 + ξ1113872 1113873 1113946

1

0vu

2tdx

α23ξ 11139461

0vwtkdxle

α23ξ2ϵ

M231 d

minus 11 d

minus (43)2 b 1113946

1

0u5dx1113888 1113889

13

+ g 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+α235ϵ

ξ2 11139461

0u5dx +

3α2310ϵ

ξ2 11139461

0v5dx +

α23ϵξ2

bdminus 11 + gd

minus 11 + ξ1113872 1113873 1113946

1

0vw

2tdx

α31ξ 11139461

0wuthdxle

α31sξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α31e5ϵ

ξ2 11139461

0u5dx

+α31ξ

2

2ϵr +

35

e1113874 1113875 11139461

0w

5dx +α31ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0wu

2tdx

α32ξ 11139461

0wvthdxle

α32sξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α32e5ϵ

ξ2 11139461

0u5dx

+α32ξ

2

2ϵr +

35

e1113874 1113875 11139461

0w

5dx +α32ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0wv

2tdx

(75)

Complexity 15

erefore

11139461

01 + p11ξ( 1113857utfdx + ξ 1113946

1

01 + p22( 1113857vtkdx + 1113946

1

0η + p33ξ( 1113857wthdx

+ ξ 11139461

0p12vtfdx + ξ 1113946

1

0p13wtfdx + ξ 1113946

1

0p21utkdx

+ ξ 11139461

0p23wtkdx + ξ 1113946

1

0p31uthdx + ξ 1113946

1

0p32vthdx

le λϵξ 11139461

0(u + v + w) u

2t + v

2t + w

2t1113872 1113873dx +

C8

ϵM0d

minus 11 d

minus 22 (1 + ξ + η)

+C9

ϵM

231 d

minus (43)2 ξ 1 + d

minus 11 + ξ + η1113872 1113873 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx1113890 1113891

13

+C10

ϵξ2 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx

(76)



12

11139461

0P2xxdx minus

ξ2

11139461

0Q

2xxdx minus

η2

11139461

0R2xxdx

+ B1K2dminus 11 d

minus 12 + B2K3d

minus (43)1 z

13


23+ B4K5z

(77)

where z 111393810(u5 + v5 + w5)dx K2 (1 + ξ + η)


2 ) K3 M231 ξ(1 + dminus 1

1 + ξ + η)



zleB5ξminus (52)

11139461

0P52

+ Q52

+ R52

1113872 1113873dx

leB6ξminus (52)

K321 d

minus (32)1 y

12+ B6ξ

minus (52)K

521 d

minus (52)1

z13 leB7ξ

minus (56)K

121 d

minus (12)1 y

16+ B7ξ

minus (56)K

561 d

minus (56)1

z23 leB8ξ

minus (53)K1d

minus 11 y

13+ B8ξ

minus (53)K

531 d

minus (53)1

(78)


431 d

minus (43)1 (|Pxx|22 + K2

1dminus 21 ) So

minusξ2

1113946

1

0

P2xxdx minus

12

1113946

1

0

Q2xxdx minus

η2

1113946

1

0

R2xxdx


431 y

53+(1 + ξ + η)K

21d

minus 21

(79)


2 we have

12


432 y

53

+ A2ξminus (56)

K121 K3d

minus (16)2 y

16

+ A3ξminus (53)

K1K4dminus (13)2 y

13

+ A4ξminus (52)

K321 K5d

minus (12)2 y

12

+ A5 K21(1 + ξ + η) + K2ξ1113960

+ K561 K3ξ

minus (56)d

minus (16)2 + K

531 K4ξ

minus (53)d

minus (12)2

+ K521 K5ξ

minus (52)d

minus (12)2 1113961

(80)



11139461

0d2Px( 1113857

2dx 11139461

0d2Qx( 1113857

2dx 11139461

0d2Rx( 1113857

2dxle 1113958M2 tge 1113957τ2

(81)




16 Complexity

ux

vx

wx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ J

minus 1

Px

Qx

Rx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

J

Pu Pv Pw

Qu Qv Qw

Ru Rv Rw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(82)


d2ux

11138681113868111386811138681113868111386811138681113868 + d2vx

11138681113868111386811138681113868111386811138681113868 + d2wx

11138681113868111386811138681113868111386811138681113868leL d2Px

11138681113868111386811138681113868111386811138681113868 + d2Qx

11138681113868111386811138681113868111386811138681113868 + d2Rx

111386811138681113868111386811138681113868111386811138681113872 1113873

0lt xlt 1 tgt 0

(83)



11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w





11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w














2d1 + 2α11M + α12M + α13M( 1113857

+ λ1113957vM2 α131113957u + ρα31 1113957w( 1113857

2d2 + α21M + 2α22M + α23M( 1113857

+ ρ1113957wM2 α121113957u + λα211113957v( 1113857

2d3 + α31M + α32M + 2α33M( 1113857

(87)



Define

V(u v w) 11139461


u


v

1113957v1113874 11138751113882


1113957w1113874 11138751113883dx

(88)


dV(u v w)

dt minus 1113946

1

0

1113957u

u2 d1 + p11( 1113857u2x +

λ1113957v

v2d2 + p22( 1113857v

2x1113896

+ρ1113957w

w2 d3 + p33( 1113857w2x +

α121113957u

u+λα211113957v

v1113888 1113889uxvx

+α131113957u

u+ρα31 1113957w

w1113874 1113875uxwx

+λα231113957v

v+ρα32 1113957w

w1113888 1113889vxwx1113897dx

minus 11139461

0

v

1113957uu(u minus 1113957u)

2+

bλu

1113957vv(v minus 1113957v)

21113896

+ ρr(w minus 1113957w)2

minusgλ + 1



(89)

Complexity 17


4λρ1113957u1113957v1113957w d1 + p11( 1113857 d2 + p22( 1113857 d3 + p33( 1113857


gt ρ1113957w α121113957uv + λα211113957vu( 11138572

d3 + p33( 1113857

+ λ1113957v α131113957uw + ρα31 1113957wu( 11138572

d2 + p22( 1113857

+ 1113957u λα231113957vw + ρα32 1113957wv( 11138572

d1 + p11( 1113857

(90)



1

0(u minus 1113957u)

2+(v minus 1113957v)

2+(w minus 1113957w)

21113960 1113961dx

dH(u v w)

dtlt 0 (u v w)neE3

(91)


10[(u minus 1113957u)2 + (v minus 1113957v)2 +





t⟶infin

(92)





t⟶infin

(93)



α11 2

α12 1

α13 10

α21 32

α22 12

α23 1

α31 10

α32 2

α33 3

a 12

b 12

g 1

s 3

e 1

r 5

d1 d2 d3≫ 1

(94)


Data Availability




18 Complexity

Acknowledgments


References



























Complexity 19

(4)







vs ws) reads

J us( 1113857

j11 j12 j13

j21 j22 j23

j31 j32 j33

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (11)


j12 1 + aus

j13 us

j21 b minus vs


j23 0

j31 minus ews

j32 0


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)



+ α2λ + α3 0 (13)

with

α1 11113957u

g1113957u + 1113957v + +1113957u2

+ r1113957u1113957w1113872 1113873gt 0

α2 11113957u


1113957w1113872 1113873

α3 11113957u


α1α2 minus α3 r1113957w1113957u2

+ (1 minus ag)1113957v + 2gr1113957w + re1113957w2

+ r2

1113957w2

1113872 11138731113957u

+ rg2

1113957w + e1113957v1113957w + 2r1113957v1113957w + gr2

1113957w2

+(1 minus ag)g1113957v +r1113957w1113957v2

1113957u2 +11113957u


+ r2

1113957w21113957v + 2gr1113957v1113957w1113872 1113873

(14)






X oplus+infin

i1Xi

Xi oplusdimE μi( )

j1Xij

(15)


Theorem 2













(16)



4 Complexity

I1 d21 d2 + d3( 1113857 + d

22 d1 + d3( 1113857 + d

23 d1 + d2( 1113857 + 2d1d2d3


231113872 1113873 minus j22 d

21 + d

231113872 1113873

minus 2 j11 + j22( 1113857 d1d2 + d2d3 + d1d3( 1113857

I3 j11 + j22( 1113857 d1j22 + d2j11 + d3j11 + d3j22( 1113857



(17)





+ Aiμ2i ξ2

+ Biμiξ +


limi⟶infin

1113957φi(ξ)

μ3i1113896 1113897 ξ3 + d1 + d2 + d3( 1113857ξ2

+ d1d2 + d2d3 + d1d3( 1113857ξ + d1d2d3 ≜ 1113957φ(ξ)

(19)





Theorem 3




A2 in [20] show




(20)


(1) Define

V(u v w) 1113946Ω



v

v1113874 11138751113876


w1113874 11138751113877dx

(21)


dV(u v w)

dt minus 1113946Ω

d1u

u2 |nablau|2

+λd2v

v2|nablav|

2+ρd3w

w2 |nablaw|2

1113888 1113889dx

minus1s

1113946Ω


s

v

u(u minus u)

2+


eg

u

v(v minus v)

2+

(e + as)

s

v

u

1113971

(u minus u) minus

s(e + as)u

v

1113970


2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭dx

le minus 1113946Ω

d1u

u2 |nablau|2

+λd2v

v2|nablav|

2+ρd3w

w2 |nablaw|2

1113888 1113889dx

minus1s

1113946Ω


s

v

u(u minus u)

2+


eg

u

v(v minus v)

21113896 1113897dx

(22)

Complexity 5


limt⟶infin

1113946Ω

|nablau|2

+|nablav|2

+|nablaw|2

1113872 1113873dx 0 (23)

limt⟶infin

1113946Ω

(u minus u)2dx 0

limt⟶infin

1113946Ω

(v minus v)2dx 0

(24)


limt⟶infin

1113946Ω

(u minus 1113954u)2dx 0

limt⟶infin

1113946Ω

(v minus 1113954v)2dx 0

limt⟶infin

1113946Ω

(w minus 1113954w)2dx 0

(25)



limt⟶infin

1113954u(t) u

limt⟶infin

1113954v(t) v(26)


1113954uprime tm( 1113857 1113946Ω



(27)


limtm⟶infin

1113954w(t) w (28)





m⟶infin

(29)



C2(Ω)⟶ 0


C2(Ω)⟶ 0


C2(Ω)⟶ 0

m⟶infin

(30)


(2) Define

V(u v w) 1113946Ω


u21113888 11138891113890


v21113888 1113889 +

1e

w1113891dx

(31)


dV(u v w)

dt minus 1113946Ω

d1u2

u2 |nablau|2

+λd2v2

v2|nablav|

21113888 1113889dx minus 1113946

Ω


v


21113896

+(1 minus ag)2 b


u


2+

r

ew

2

+1

b minus g

v

u

1113970


u

v

1113970

v minus v2( 11138571113888 1113889

2⎫⎬

⎭dx

(32)




6 Complexity



zuzt


zuz]

0 x isin zΩ tgt 0


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(33)


P

p11 p12 p13

p21 p22 p23

p31 p33 p33

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (34)

with

p11 2α11us + α12vs + α13ws

p12 α12us

p13 α13us

p21 α21vs

p22 α21us + 2α22vs + α23ws

p23 α23vs

p31 α31ws

p32 α32ws

p33 α31us + α32vs + 2α33ws

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(35)

We denote

L ≔ minus μi(D + P) + J (36)


ϕ λ3 + c1λ2

+ c2λ + c3 (37)

where




2i + β6μi + β7

(38)

with



β3 M22 + M33 + j22j33








(39)








Corollary 1



Complexity 7




wα23 + d2)


eg +


1113969







1113969







β1α13

21113957w3α23α33 + 21113957uα21α33 + 41113957vα22α33 + 2d3α33 + d3α23( 1113857 1113957w

2



β2α13




β3α13

(s minus 2e1113957u)(g + 1113957u)

(40)

Note that

limα13⟶infin

C α13 ]i( 1113857

α13 μi C1μ

2i + C2μi + C31113872 1113873 (41)


eg +


1113969


limα13⟶infin


C2 minus 4C1C3

1113968

2C1lt 0

limα13⟶infin

1113957μ2 0

limα13⟶infin

1113957μ3 minus C2 +

C2 minus 4C1C3

1113968

2C1gt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(42)





⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(43)







(1) Let

8 Complexity

α11 α12 α21 α22 α23 α32 1

α13 12

α31 3

α33 4

a 05

b 1

g 15

s 3

e 2

r 4

d1 3

d2 3

d3 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(44)


0326 0388) c1 5809gt 0 c2 10163gt 0 c3


(2) Let

α11 α12 α21 α22 α23 α31 α32 α33 1

d1 3

d2 3

d3 2

a 05

b 13

g 15

s 3

e 5

r 4

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(45)



(3) Let αij 01 i j 1 2 3 a 28 b 1 g 3 s










(46)






|u|2 leC ux

11138681113868111386811138681113868111386811138681113868132 |u|

231 + |u|11113874 1113875 forallu isinW

12(0 1) (47)

|u|4 leC ux

11138681113868111386811138681113868111386811138681113868122 |u|

121 + |u|11113874 1113875 forallu isinW

12(0 1) (48)

|u|52 leC ux

11138681113868111386811138681113868111386811138681113868252 |u|

351 + |u|11113874 1113875 forallu isinW

12(0 1) (49)

ux

111386811138681113868111386811138681113868111386811138682leC uxx

11138681113868111386811138681113868111386811138681113868352 |u|

251 + |u|11113874 1113875 forallu isinW

22(0 1) (50)



(H1) ablt 1 bltg




(51)


(52)


Complexity 9








ddt

11139461

0u + ρv +

1e

w1113874 1113875dxleC1 minus l 11139461

0u + ρv +

1e

w1113874 1113875dx

(53)


11139461

0udx 1113946

1

0vdx 1113946

1



11139461

0udx 1113946

1

0vdx 1113946

1



12ddt

11139461


1


1

0p11u

2xdx minus 1113946

1


1

0p13uxwxdx + 1113946

1

0uv + au

2v + u

2w1113872 1113873dx

12ddt

11139461


1


1

0p22v

2xdx minus 1113946

1


1

0p23vxwxdx + b 1113946

1

0uvdx

12ddt

11139461

0w


0w

2xdx minus 1113946

1

0p33w

2xdx minus 1113946

1


1

0p32vxwxdx + s 1113946

1

0w

2dx

(56)


12ddt

11139461

0u2

+ v2

+ w2


11139461

0u2x + v

2x + w

2x1113872 1113873dx minus 1113946

1


+(b + 1) 11139461

0uvdx + a 1113946

1

0u2vdx + s 1113946

1

0w

2dx

(57)

where

q ux vx wx( 1113857 p11u2x + p22v

2x + p33w

2x + p12 + p21( 1113857uxvx

+ p23 + p32( 1113857vxwx + p13 + p31( 1113857uxwx

(58)


12ddt

11139461

0u2

+ v2

+ w2


11139461

0u2x + v

2x + w

2x1113872 1113873dx

+(b + 1) 11139461

0uvdx

+ a 11139461

0u2vdx + s 1113946

1

0w

2dx

(59)

10 Complexity



|u|41)leCM20(|ux|22 + M2


a 11139461

0u2vdxle a 1113946

1

0

u4

2εdx + a 1113946

1

0

ε2v2dx

a2CM2

02dlowast

11139461

0v2dx +

dlowast

211139461

0u2xdx +

dlowastM20

2

(60)


11139461

0u2

+ v2

+ w2


211139461

0u2x + v

2x + w

2x1113872 1113873dx

+ C2 11139461

0u2

+ v2

+ w2

1113872 1113873dx +dlowastM2

02

(61)


|u|61 leCM40(|ux|22 + M2

0) erefore

minus 11139461

0u2x + v

2x + w

2x1113872 1113873dxle 3M

20 minus

19CM2

011139461

0u2

+ v2

+ w2

1113872 1113873dx1113890 1113891

3

(62)


12ddt

11139461

0u2

+ v2

+ w2


11139461

0u2

+ v2

+ w2

1113872 1113873dx1113890 1113891

3

+ C2 11139461

0u2

+ v2

+ w2

1113872 1113873dx + 2dlowastM

20

(63)


such that

11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w





11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w



1113957u u

d2

1113957v v

d2

1113957w w

d2

1113957t d1t

(66)






u(x 0) u0(x)

v(x 0) v0(x)

w(x 0) w0(x)

0ltxlt 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(67)




1u minus gdminus 1




11139461

0udx 1113946

1

0vdx 1113946

1

0wdxleM0d

minus 12

11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w

2dxleM1dminus 22


(68)


1


1

0v2tdx

minus η11139461

0w


1


+ 11139461


+ ξ11139461

0p21utk + 1+ p22( 1113857vtk + p23wtk1113858 1113859dx

+ 11139461


(69)

Complexity 11

where y 111393810(P2

x + Q2x + R2


1 2 3) such that


2t + w

2t1113872 1113873 (70)

erefore


1


1


1

0w


1

0(u + v + w) u

2t + v

2t + w

2t1113872 1113873dx

+ 11139461

01 + p11ξ( 1113857utfdx + 1113946

1

0ξ 1 + p22( 1113857vtkdx + 1113946

1

0η + p33ξ( 1113857wthdx

+ 11139461

0p12ξvtfdx + 1113946

1

0p13ξwtfdx + 1113946

1

0p21ξutkdx

+ 11139461

0p23ξwtkdx + 1113946

1

0p31ξuthdx + 1113946

1

0p32ξvthdx

(71)


11139461

0u4dxleM

131 d

minus (23)1 1113946

1

0u5dx1113888 1113889

23

11139461

0u2v2dxleM

131 d

minus (23)1 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

11139461

0u3dxleM

231 d

minus (43)1 1113946

1

0u5dx1113888 1113889

13

11139461

0uv

2dxleM231 d

minus (43)1 1113946

1

0v5dx1113888 1113889

13

11139461

0u3wdxleM

131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

12

11139461

0w

5dx1113888 1113889

16

11139461

0uvwdxleM

231 d

minus (43)2 1113946

1

0u5dx1113888 1113889

16

11139461

0v5dx1113888 1113889

16

11139461

0u2vwdxleM

131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

16

(72)



12 Complexity

minus 11139461

0u2tdxle minus

12

11139461

0P2xxdx + 1113946

1

0f2dx

minus ξ 11139461

0v2tdxle minus

ξ2

11139461

0Q

2xxdx + ξ 1113946

1

0k2dx

minus η11139461

0w

2tdxle minus

η2

11139461

0R2xxdx + η1113946

1

0h2dx

11139461

0f2dxle 2M1d

minus 21 d

minus 22 + a

2ξ2M131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

+ ξ2M131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0w

5dx1113888 1113889

13

+ 2aξdminus 11 d

minus (43)2 M

231 1113946

1

0v5dx1113888 1113889

13

+ 2ξdminus 11 d

minus (23)2 M

231 1113946

1

0u5dx1113888 1113889

16

11139461

0v5dx1113888 1113889

16


131 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

16

ξ 11139461

0k2dxle ξd

minus 21 d

minus 22 M1 b

2+ g

21113872 1113873 + ξ3dminus (23)

2 M131 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

+ ξ2gdminus 11 d

minus (43)2 M

231 1113946

1

0v5dx1113888 1113889

13

η11139461

0h2dxle ηs

2d

minus 21 d

minus 22 M1 + ηd

2ξ2dminus (23)2 M

131 1113946

1

0u5dx1113888 1113889

13

ηr2ξ2dminus (23)

2 M131


2 M131 1113946

1

0u5dx1113888 1113889

23


131 1113946

1

0u5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

12

(73)

us

minus 11139461


1


1

0w

2tdxle minus

12

11139461

0P2xxdx minus

ξ2

11139461

0Q

2xxdx minus

η2

11139461

0R2xxdx + C5(1 + ξ + η)d

minus 21 d

minus 22 M1

+ C6dminus (43)2 M

231 ξ(1 + η) 1113946

1

0u5

+ v5

1113872 1113873dx1113888 1113889

13

+ C7dminus (23)2 M

131 ξ2(1 + η) 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx1113888 1113889

23

(74)

Complexity 13

11139461

0utfdxle d

minus 11 1113946

1

0uutdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ d

minus 11 1113946

1

0utvdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ aξ 1113946

1

0uutvdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ ξ 1113946

1

0uutwdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le dminus 11

12ϵ

11139461

0udx +ϵ2

11139461

0uu

2tdx1113888 1113889 + d

minus 11

12ϵ

11139461

0vdx +ϵ2

11139461

0vu

2tdx1113888 1113889

+ aξ12ϵ

11139461

0uv

2dx +ϵ2

11139461

0uu

2tdx1113888 1113889 + ξ

12ϵ

11139461

0uw

2dx +ϵ2

11139461

0uu

2tdx1113888 1113889


2ϵ

M0 +a

2ϵM

231 d

minus (43)2 ξ 1113946

1

0v5dx1113888 1113889

13

+ξ2ϵ

M231 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+ϵdminus 1

12

11139461

0vu

2tdx +ϵ2

dminus 11 + aξ + ξ1113872 1113873 1113946

1

0uu

2tdx

2α11ξ 11139461

0uutfdxle

α11ϵ

M231 d

minus 11 d

232 1113946

1

0u5dx1113888 1113889

13

+ 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦ +

3α11(a + 1)

5ϵξ2 1113946

1

0u5dx

+2α11a5ϵ

ξ2 11139461

0v5dx +

2α115ϵ

ξ2 11139461

0w

5dx + α11ξϵ 2dminus 11 + aξ + ξ1113872 1113873 1113946

1

0uu

2tdx

2α12ξ 11139461

0vutfdxle

α12ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0v5dx1113888 1113889

13

+α12(a + 1)

5ϵξ2 1113946

1

0u5dx +

2α125ϵ

ξ2 11139461

0w

5dx +α12(3a + 4)

10ϵξ2 1113946

1

0v5dx

α13ξ 11139461

0wutfdxle

α13ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α13(a +(12))

5ϵξ2 1113946

1

0u5dx +

2aα135ϵ

ξ2 11139461

0v5dx

+2α13(a + 1)

5ϵξ2 1113946

1

0w

5dx +α13dminus 1

12ϵξ 1113946

1

0vu

2tdx +ϵ2


2+ 11113872 1113873 1113946

1

0uu

2tdx

ξ 11139461

0vtkdxle

b + g

2ϵM0d

minus 11 d

minus 12 ξ +

ξ2

2ϵM

231 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+bdminus 1

12ϵξ 1113946

1

0uv

2tdx +ϵ2ξ gd

minus 11 + ξ1113872 1113873 1113946

1

0vv

2tdx

α21ξ 11139461

0uvtkdxle

α21(b + g)

2ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+2α215ϵ

ξ2 11139461

0u5dx

+α2110ϵ

ξ2 11139461

0v5dx +

α21bdminus 11

2ϵξ 1113946

1

0uv

2tdx +

α21ϵξ2

gdminus 11 + ξ1113872 1113873 1113946

1

0vv

2tdx

2α22ξ 11139461

0vvtk dx le

α22(b + g)

ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+α225ϵ

ξ2 11139461

0u5dx

+4α225ϵ

ξ2 11139461

0v5dx + α22(b + g)d

minus 11 ϵξ 1113946

1

0vv

2tdx + α22ϵξ

211139461

0uv

2tdx

α23ξ 11139461

0wvtkdxle

α23(b + g)

2ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

minus (43)

+α23ϵξ2

bdminus 11 + ξ1113872 1113873 1113946

1

0uv

2tdx

+α235ϵ

ξ2 11139461

0u5dx + 21113946

1

0v5dx + 21113946

1

0w

5dx1113888 1113889 +α23gdminus 1

12ϵξ 1113946

1

0vv

2tdx

η11139461

0wthdxle

s

ϵd

minus 11 d

minus 12 M0 +

r

2ϵM

231 d

minus (43)2 ηξ 1113946

1

0w

5dx1113888 1113889

13

+e

2ϵM

231 d

minus (43)2 ηξ 1113946

1

0w

5dx1113888 1113889

13ηϵ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

α31ξ 11139461

0uwthdxle

α31s2ϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0u5dx1113888 1113889

13

+α31(r + 2e)

5ϵξ2 1113946

1

0u5dx1113888 1113889

13

+α31(3r + e)

10ϵξ2 1113946

1

0w

5dx1113888 1113889

13

+α31ξϵ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

14 Complexity

α32ξ 11139461

0vwthdxle

α32s2ϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0v5dx1113888 1113889

13

+2α32e5ϵ

ξ2 11139461

0u5dx +

α32(r + 2e)

5ϵξ2 1113946

1

0v5dx

+α32(e +(3r2))

5ϵξ2 1113946

1

0w

5dx +α32ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

2α33ξ 11139461

0wwthdxle

α33sϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0w

5dx1113888 1113889

13

+2α33e5ϵ

ξ2

+α33(5r + 3e)

5ϵξ2 1113946

1

0w


1

0ww

2tdx

α12ξ 11139461

0uvtfdxle

α12ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+α125ϵ

2a +32

1113874 1113875ξ2 11139461

0u5dx

+aα1210ϵ

ξ2 11139461

0v5dx +

α125ϵ

ξ2 11139461

0w

5dx +α12ϵξ2

dminus 11 + ξ1113872 1113873 1113946

1

0uv

2tdx +

α12ϵξ2

dminus 11 + aξ1113872 1113873 1113946

1

0vv

2tdx

α12ξ 11139461

0uwtfdxle

α12ξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+ 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+3α12(a + 1)

10ϵξ2 1113946

1

0u5dx +

aα125ϵ

ξ2 11139461

0v5dx +

α125ϵ

ξ2 11139461

0w


aξ2

+ξ2

1113888 1113889 11139461

0uw

2tdx

α21ξ 11139461

0vutkdxle

α21ξ2ϵ

M231 d

minus 11 d

minus (43)2 b 1113946

1

0u5dx1113888 1113889

13

+ g 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+α215ϵ

ξ2 11139461

0u5dx +

3α2110ϵ

ξ2 11139461

0v5dx +

α21ϵξ2

bdminus 11 + gd

minus 11 + ξ1113872 1113873 1113946

1

0vu

2tdx

α23ξ 11139461

0vwtkdxle

α23ξ2ϵ

M231 d

minus 11 d

minus (43)2 b 1113946

1

0u5dx1113888 1113889

13

+ g 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+α235ϵ

ξ2 11139461

0u5dx +

3α2310ϵ

ξ2 11139461

0v5dx +

α23ϵξ2

bdminus 11 + gd

minus 11 + ξ1113872 1113873 1113946

1

0vw

2tdx

α31ξ 11139461

0wuthdxle

α31sξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α31e5ϵ

ξ2 11139461

0u5dx

+α31ξ

2

2ϵr +

35

e1113874 1113875 11139461

0w

5dx +α31ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0wu

2tdx

α32ξ 11139461

0wvthdxle

α32sξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α32e5ϵ

ξ2 11139461

0u5dx

+α32ξ

2

2ϵr +

35

e1113874 1113875 11139461

0w

5dx +α32ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0wv

2tdx

(75)

Complexity 15

erefore

11139461

01 + p11ξ( 1113857utfdx + ξ 1113946

1

01 + p22( 1113857vtkdx + 1113946

1

0η + p33ξ( 1113857wthdx

+ ξ 11139461

0p12vtfdx + ξ 1113946

1

0p13wtfdx + ξ 1113946

1

0p21utkdx

+ ξ 11139461

0p23wtkdx + ξ 1113946

1

0p31uthdx + ξ 1113946

1

0p32vthdx

le λϵξ 11139461

0(u + v + w) u

2t + v

2t + w

2t1113872 1113873dx +

C8

ϵM0d

minus 11 d

minus 22 (1 + ξ + η)

+C9

ϵM

231 d

minus (43)2 ξ 1 + d

minus 11 + ξ + η1113872 1113873 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx1113890 1113891

13

+C10

ϵξ2 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx

(76)



12

11139461

0P2xxdx minus

ξ2

11139461

0Q

2xxdx minus

η2

11139461

0R2xxdx

+ B1K2dminus 11 d

minus 12 + B2K3d

minus (43)1 z

13


23+ B4K5z

(77)

where z 111393810(u5 + v5 + w5)dx K2 (1 + ξ + η)


2 ) K3 M231 ξ(1 + dminus 1

1 + ξ + η)



zleB5ξminus (52)

11139461

0P52

+ Q52

+ R52

1113872 1113873dx

leB6ξminus (52)

K321 d

minus (32)1 y

12+ B6ξ

minus (52)K

521 d

minus (52)1

z13 leB7ξ

minus (56)K

121 d

minus (12)1 y

16+ B7ξ

minus (56)K

561 d

minus (56)1

z23 leB8ξ

minus (53)K1d

minus 11 y

13+ B8ξ

minus (53)K

531 d

minus (53)1

(78)


431 d

minus (43)1 (|Pxx|22 + K2

1dminus 21 ) So

minusξ2

1113946

1

0

P2xxdx minus

12

1113946

1

0

Q2xxdx minus

η2

1113946

1

0

R2xxdx


431 y

53+(1 + ξ + η)K

21d

minus 21

(79)


2 we have

12


432 y

53

+ A2ξminus (56)

K121 K3d

minus (16)2 y

16

+ A3ξminus (53)

K1K4dminus (13)2 y

13

+ A4ξminus (52)

K321 K5d

minus (12)2 y

12

+ A5 K21(1 + ξ + η) + K2ξ1113960

+ K561 K3ξ

minus (56)d

minus (16)2 + K

531 K4ξ

minus (53)d

minus (12)2

+ K521 K5ξ

minus (52)d

minus (12)2 1113961

(80)



11139461

0d2Px( 1113857

2dx 11139461

0d2Qx( 1113857

2dx 11139461

0d2Rx( 1113857

2dxle 1113958M2 tge 1113957τ2

(81)




16 Complexity

ux

vx

wx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ J

minus 1

Px

Qx

Rx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

J

Pu Pv Pw

Qu Qv Qw

Ru Rv Rw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(82)


d2ux

11138681113868111386811138681113868111386811138681113868 + d2vx

11138681113868111386811138681113868111386811138681113868 + d2wx

11138681113868111386811138681113868111386811138681113868leL d2Px

11138681113868111386811138681113868111386811138681113868 + d2Qx

11138681113868111386811138681113868111386811138681113868 + d2Rx

111386811138681113868111386811138681113868111386811138681113872 1113873

0lt xlt 1 tgt 0

(83)



11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w





11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w














2d1 + 2α11M + α12M + α13M( 1113857

+ λ1113957vM2 α131113957u + ρα31 1113957w( 1113857

2d2 + α21M + 2α22M + α23M( 1113857

+ ρ1113957wM2 α121113957u + λα211113957v( 1113857

2d3 + α31M + α32M + 2α33M( 1113857

(87)



Define

V(u v w) 11139461


u


v

1113957v1113874 11138751113882


1113957w1113874 11138751113883dx

(88)


dV(u v w)

dt minus 1113946

1

0

1113957u

u2 d1 + p11( 1113857u2x +

λ1113957v

v2d2 + p22( 1113857v

2x1113896

+ρ1113957w

w2 d3 + p33( 1113857w2x +

α121113957u

u+λα211113957v

v1113888 1113889uxvx

+α131113957u

u+ρα31 1113957w

w1113874 1113875uxwx

+λα231113957v

v+ρα32 1113957w

w1113888 1113889vxwx1113897dx

minus 11139461

0

v

1113957uu(u minus 1113957u)

2+

bλu

1113957vv(v minus 1113957v)

21113896

+ ρr(w minus 1113957w)2

minusgλ + 1



(89)

Complexity 17


4λρ1113957u1113957v1113957w d1 + p11( 1113857 d2 + p22( 1113857 d3 + p33( 1113857


gt ρ1113957w α121113957uv + λα211113957vu( 11138572

d3 + p33( 1113857

+ λ1113957v α131113957uw + ρα31 1113957wu( 11138572

d2 + p22( 1113857

+ 1113957u λα231113957vw + ρα32 1113957wv( 11138572

d1 + p11( 1113857

(90)



1

0(u minus 1113957u)

2+(v minus 1113957v)

2+(w minus 1113957w)

21113960 1113961dx

dH(u v w)

dtlt 0 (u v w)neE3

(91)


10[(u minus 1113957u)2 + (v minus 1113957v)2 +





t⟶infin

(92)





t⟶infin

(93)



α11 2

α12 1

α13 10

α21 32

α22 12

α23 1

α31 10

α32 2

α33 3

a 12

b 12

g 1

s 3

e 1

r 5

d1 d2 d3≫ 1

(94)


Data Availability




18 Complexity

Acknowledgments


References



























Complexity 19

I1 d21 d2 + d3( 1113857 + d

22 d1 + d3( 1113857 + d

23 d1 + d2( 1113857 + 2d1d2d3


231113872 1113873 minus j22 d

21 + d

231113872 1113873

minus 2 j11 + j22( 1113857 d1d2 + d2d3 + d1d3( 1113857

I3 j11 + j22( 1113857 d1j22 + d2j11 + d3j11 + d3j22( 1113857



(17)





+ Aiμ2i ξ2

+ Biμiξ +


limi⟶infin

1113957φi(ξ)

μ3i1113896 1113897 ξ3 + d1 + d2 + d3( 1113857ξ2

+ d1d2 + d2d3 + d1d3( 1113857ξ + d1d2d3 ≜ 1113957φ(ξ)

(19)





Theorem 3




A2 in [20] show




(20)


(1) Define

V(u v w) 1113946Ω



v

v1113874 11138751113876


w1113874 11138751113877dx

(21)


dV(u v w)

dt minus 1113946Ω

d1u

u2 |nablau|2

+λd2v

v2|nablav|

2+ρd3w

w2 |nablaw|2

1113888 1113889dx

minus1s

1113946Ω


s

v

u(u minus u)

2+


eg

u

v(v minus v)

2+

(e + as)

s

v

u

1113971

(u minus u) minus

s(e + as)u

v

1113970


2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭dx

le minus 1113946Ω

d1u

u2 |nablau|2

+λd2v

v2|nablav|

2+ρd3w

w2 |nablaw|2

1113888 1113889dx

minus1s

1113946Ω


s

v

u(u minus u)

2+


eg

u

v(v minus v)

21113896 1113897dx

(22)

Complexity 5


limt⟶infin

1113946Ω

|nablau|2

+|nablav|2

+|nablaw|2

1113872 1113873dx 0 (23)

limt⟶infin

1113946Ω

(u minus u)2dx 0

limt⟶infin

1113946Ω

(v minus v)2dx 0

(24)


limt⟶infin

1113946Ω

(u minus 1113954u)2dx 0

limt⟶infin

1113946Ω

(v minus 1113954v)2dx 0

limt⟶infin

1113946Ω

(w minus 1113954w)2dx 0

(25)



limt⟶infin

1113954u(t) u

limt⟶infin

1113954v(t) v(26)


1113954uprime tm( 1113857 1113946Ω



(27)


limtm⟶infin

1113954w(t) w (28)





m⟶infin

(29)



C2(Ω)⟶ 0


C2(Ω)⟶ 0


C2(Ω)⟶ 0

m⟶infin

(30)


(2) Define

V(u v w) 1113946Ω


u21113888 11138891113890


v21113888 1113889 +

1e

w1113891dx

(31)


dV(u v w)

dt minus 1113946Ω

d1u2

u2 |nablau|2

+λd2v2

v2|nablav|

21113888 1113889dx minus 1113946

Ω


v


21113896

+(1 minus ag)2 b


u


2+

r

ew

2

+1

b minus g

v

u

1113970


u

v

1113970

v minus v2( 11138571113888 1113889

2⎫⎬

⎭dx

(32)




6 Complexity



zuzt


zuz]

0 x isin zΩ tgt 0


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(33)


P

p11 p12 p13

p21 p22 p23

p31 p33 p33

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (34)

with

p11 2α11us + α12vs + α13ws

p12 α12us

p13 α13us

p21 α21vs

p22 α21us + 2α22vs + α23ws

p23 α23vs

p31 α31ws

p32 α32ws

p33 α31us + α32vs + 2α33ws

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(35)

We denote

L ≔ minus μi(D + P) + J (36)


ϕ λ3 + c1λ2

+ c2λ + c3 (37)

where




2i + β6μi + β7

(38)

with



β3 M22 + M33 + j22j33








(39)








Corollary 1



Complexity 7




wα23 + d2)


eg +


1113969







1113969







β1α13

21113957w3α23α33 + 21113957uα21α33 + 41113957vα22α33 + 2d3α33 + d3α23( 1113857 1113957w

2



β2α13




β3α13

(s minus 2e1113957u)(g + 1113957u)

(40)

Note that

limα13⟶infin

C α13 ]i( 1113857

α13 μi C1μ

2i + C2μi + C31113872 1113873 (41)


eg +


1113969


limα13⟶infin


C2 minus 4C1C3

1113968

2C1lt 0

limα13⟶infin

1113957μ2 0

limα13⟶infin

1113957μ3 minus C2 +

C2 minus 4C1C3

1113968

2C1gt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(42)





⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(43)







(1) Let

8 Complexity

α11 α12 α21 α22 α23 α32 1

α13 12

α31 3

α33 4

a 05

b 1

g 15

s 3

e 2

r 4

d1 3

d2 3

d3 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(44)


0326 0388) c1 5809gt 0 c2 10163gt 0 c3


(2) Let

α11 α12 α21 α22 α23 α31 α32 α33 1

d1 3

d2 3

d3 2

a 05

b 13

g 15

s 3

e 5

r 4

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(45)



(3) Let αij 01 i j 1 2 3 a 28 b 1 g 3 s










(46)






|u|2 leC ux

11138681113868111386811138681113868111386811138681113868132 |u|

231 + |u|11113874 1113875 forallu isinW

12(0 1) (47)

|u|4 leC ux

11138681113868111386811138681113868111386811138681113868122 |u|

121 + |u|11113874 1113875 forallu isinW

12(0 1) (48)

|u|52 leC ux

11138681113868111386811138681113868111386811138681113868252 |u|

351 + |u|11113874 1113875 forallu isinW

12(0 1) (49)

ux

111386811138681113868111386811138681113868111386811138682leC uxx

11138681113868111386811138681113868111386811138681113868352 |u|

251 + |u|11113874 1113875 forallu isinW

22(0 1) (50)



(H1) ablt 1 bltg




(51)


(52)


Complexity 9








ddt

11139461

0u + ρv +

1e

w1113874 1113875dxleC1 minus l 11139461

0u + ρv +

1e

w1113874 1113875dx

(53)


11139461

0udx 1113946

1

0vdx 1113946

1



11139461

0udx 1113946

1

0vdx 1113946

1



12ddt

11139461


1


1

0p11u

2xdx minus 1113946

1


1

0p13uxwxdx + 1113946

1

0uv + au

2v + u

2w1113872 1113873dx

12ddt

11139461


1


1

0p22v

2xdx minus 1113946

1


1

0p23vxwxdx + b 1113946

1

0uvdx

12ddt

11139461

0w


0w

2xdx minus 1113946

1

0p33w

2xdx minus 1113946

1


1

0p32vxwxdx + s 1113946

1

0w

2dx

(56)


12ddt

11139461

0u2

+ v2

+ w2


11139461

0u2x + v

2x + w

2x1113872 1113873dx minus 1113946

1


+(b + 1) 11139461

0uvdx + a 1113946

1

0u2vdx + s 1113946

1

0w

2dx

(57)

where

q ux vx wx( 1113857 p11u2x + p22v

2x + p33w

2x + p12 + p21( 1113857uxvx

+ p23 + p32( 1113857vxwx + p13 + p31( 1113857uxwx

(58)


12ddt

11139461

0u2

+ v2

+ w2


11139461

0u2x + v

2x + w

2x1113872 1113873dx

+(b + 1) 11139461

0uvdx

+ a 11139461

0u2vdx + s 1113946

1

0w

2dx

(59)

10 Complexity



|u|41)leCM20(|ux|22 + M2


a 11139461

0u2vdxle a 1113946

1

0

u4

2εdx + a 1113946

1

0

ε2v2dx

a2CM2

02dlowast

11139461

0v2dx +

dlowast

211139461

0u2xdx +

dlowastM20

2

(60)


11139461

0u2

+ v2

+ w2


211139461

0u2x + v

2x + w

2x1113872 1113873dx

+ C2 11139461

0u2

+ v2

+ w2

1113872 1113873dx +dlowastM2

02

(61)


|u|61 leCM40(|ux|22 + M2

0) erefore

minus 11139461

0u2x + v

2x + w

2x1113872 1113873dxle 3M

20 minus

19CM2

011139461

0u2

+ v2

+ w2

1113872 1113873dx1113890 1113891

3

(62)


12ddt

11139461

0u2

+ v2

+ w2


11139461

0u2

+ v2

+ w2

1113872 1113873dx1113890 1113891

3

+ C2 11139461

0u2

+ v2

+ w2

1113872 1113873dx + 2dlowastM

20

(63)


such that

11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w





11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w



1113957u u

d2

1113957v v

d2

1113957w w

d2

1113957t d1t

(66)






u(x 0) u0(x)

v(x 0) v0(x)

w(x 0) w0(x)

0ltxlt 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(67)




1u minus gdminus 1




11139461

0udx 1113946

1

0vdx 1113946

1

0wdxleM0d

minus 12

11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w

2dxleM1dminus 22


(68)


1


1

0v2tdx

minus η11139461

0w


1


+ 11139461


+ ξ11139461

0p21utk + 1+ p22( 1113857vtk + p23wtk1113858 1113859dx

+ 11139461


(69)

Complexity 11

where y 111393810(P2

x + Q2x + R2


1 2 3) such that


2t + w

2t1113872 1113873 (70)

erefore


1


1


1

0w


1

0(u + v + w) u

2t + v

2t + w

2t1113872 1113873dx

+ 11139461

01 + p11ξ( 1113857utfdx + 1113946

1

0ξ 1 + p22( 1113857vtkdx + 1113946

1

0η + p33ξ( 1113857wthdx

+ 11139461

0p12ξvtfdx + 1113946

1

0p13ξwtfdx + 1113946

1

0p21ξutkdx

+ 11139461

0p23ξwtkdx + 1113946

1

0p31ξuthdx + 1113946

1

0p32ξvthdx

(71)


11139461

0u4dxleM

131 d

minus (23)1 1113946

1

0u5dx1113888 1113889

23

11139461

0u2v2dxleM

131 d

minus (23)1 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

11139461

0u3dxleM

231 d

minus (43)1 1113946

1

0u5dx1113888 1113889

13

11139461

0uv

2dxleM231 d

minus (43)1 1113946

1

0v5dx1113888 1113889

13

11139461

0u3wdxleM

131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

12

11139461

0w

5dx1113888 1113889

16

11139461

0uvwdxleM

231 d

minus (43)2 1113946

1

0u5dx1113888 1113889

16

11139461

0v5dx1113888 1113889

16

11139461

0u2vwdxleM

131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

16

(72)



12 Complexity

minus 11139461

0u2tdxle minus

12

11139461

0P2xxdx + 1113946

1

0f2dx

minus ξ 11139461

0v2tdxle minus

ξ2

11139461

0Q

2xxdx + ξ 1113946

1

0k2dx

minus η11139461

0w

2tdxle minus

η2

11139461

0R2xxdx + η1113946

1

0h2dx

11139461

0f2dxle 2M1d

minus 21 d

minus 22 + a

2ξ2M131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

+ ξ2M131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0w

5dx1113888 1113889

13

+ 2aξdminus 11 d

minus (43)2 M

231 1113946

1

0v5dx1113888 1113889

13

+ 2ξdminus 11 d

minus (23)2 M

231 1113946

1

0u5dx1113888 1113889

16

11139461

0v5dx1113888 1113889

16


131 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

16

ξ 11139461

0k2dxle ξd

minus 21 d

minus 22 M1 b

2+ g

21113872 1113873 + ξ3dminus (23)

2 M131 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

+ ξ2gdminus 11 d

minus (43)2 M

231 1113946

1

0v5dx1113888 1113889

13

η11139461

0h2dxle ηs

2d

minus 21 d

minus 22 M1 + ηd

2ξ2dminus (23)2 M

131 1113946

1

0u5dx1113888 1113889

13

ηr2ξ2dminus (23)

2 M131


2 M131 1113946

1

0u5dx1113888 1113889

23


131 1113946

1

0u5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

12

(73)

us

minus 11139461


1


1

0w

2tdxle minus

12

11139461

0P2xxdx minus

ξ2

11139461

0Q

2xxdx minus

η2

11139461

0R2xxdx + C5(1 + ξ + η)d

minus 21 d

minus 22 M1

+ C6dminus (43)2 M

231 ξ(1 + η) 1113946

1

0u5

+ v5

1113872 1113873dx1113888 1113889

13

+ C7dminus (23)2 M

131 ξ2(1 + η) 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx1113888 1113889

23

(74)

Complexity 13

11139461

0utfdxle d

minus 11 1113946

1

0uutdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ d

minus 11 1113946

1

0utvdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ aξ 1113946

1

0uutvdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ ξ 1113946

1

0uutwdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le dminus 11

12ϵ

11139461

0udx +ϵ2

11139461

0uu

2tdx1113888 1113889 + d

minus 11

12ϵ

11139461

0vdx +ϵ2

11139461

0vu

2tdx1113888 1113889

+ aξ12ϵ

11139461

0uv

2dx +ϵ2

11139461

0uu

2tdx1113888 1113889 + ξ

12ϵ

11139461

0uw

2dx +ϵ2

11139461

0uu

2tdx1113888 1113889


2ϵ

M0 +a

2ϵM

231 d

minus (43)2 ξ 1113946

1

0v5dx1113888 1113889

13

+ξ2ϵ

M231 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+ϵdminus 1

12

11139461

0vu

2tdx +ϵ2

dminus 11 + aξ + ξ1113872 1113873 1113946

1

0uu

2tdx

2α11ξ 11139461

0uutfdxle

α11ϵ

M231 d

minus 11 d

232 1113946

1

0u5dx1113888 1113889

13

+ 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦ +

3α11(a + 1)

5ϵξ2 1113946

1

0u5dx

+2α11a5ϵ

ξ2 11139461

0v5dx +

2α115ϵ

ξ2 11139461

0w

5dx + α11ξϵ 2dminus 11 + aξ + ξ1113872 1113873 1113946

1

0uu

2tdx

2α12ξ 11139461

0vutfdxle

α12ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0v5dx1113888 1113889

13

+α12(a + 1)

5ϵξ2 1113946

1

0u5dx +

2α125ϵ

ξ2 11139461

0w

5dx +α12(3a + 4)

10ϵξ2 1113946

1

0v5dx

α13ξ 11139461

0wutfdxle

α13ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α13(a +(12))

5ϵξ2 1113946

1

0u5dx +

2aα135ϵ

ξ2 11139461

0v5dx

+2α13(a + 1)

5ϵξ2 1113946

1

0w

5dx +α13dminus 1

12ϵξ 1113946

1

0vu

2tdx +ϵ2


2+ 11113872 1113873 1113946

1

0uu

2tdx

ξ 11139461

0vtkdxle

b + g

2ϵM0d

minus 11 d

minus 12 ξ +

ξ2

2ϵM

231 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+bdminus 1

12ϵξ 1113946

1

0uv

2tdx +ϵ2ξ gd

minus 11 + ξ1113872 1113873 1113946

1

0vv

2tdx

α21ξ 11139461

0uvtkdxle

α21(b + g)

2ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+2α215ϵ

ξ2 11139461

0u5dx

+α2110ϵ

ξ2 11139461

0v5dx +

α21bdminus 11

2ϵξ 1113946

1

0uv

2tdx +

α21ϵξ2

gdminus 11 + ξ1113872 1113873 1113946

1

0vv

2tdx

2α22ξ 11139461

0vvtk dx le

α22(b + g)

ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+α225ϵ

ξ2 11139461

0u5dx

+4α225ϵ

ξ2 11139461

0v5dx + α22(b + g)d

minus 11 ϵξ 1113946

1

0vv

2tdx + α22ϵξ

211139461

0uv

2tdx

α23ξ 11139461

0wvtkdxle

α23(b + g)

2ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

minus (43)

+α23ϵξ2

bdminus 11 + ξ1113872 1113873 1113946

1

0uv

2tdx

+α235ϵ

ξ2 11139461

0u5dx + 21113946

1

0v5dx + 21113946

1

0w

5dx1113888 1113889 +α23gdminus 1

12ϵξ 1113946

1

0vv

2tdx

η11139461

0wthdxle

s

ϵd

minus 11 d

minus 12 M0 +

r

2ϵM

231 d

minus (43)2 ηξ 1113946

1

0w

5dx1113888 1113889

13

+e

2ϵM

231 d

minus (43)2 ηξ 1113946

1

0w

5dx1113888 1113889

13ηϵ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

α31ξ 11139461

0uwthdxle

α31s2ϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0u5dx1113888 1113889

13

+α31(r + 2e)

5ϵξ2 1113946

1

0u5dx1113888 1113889

13

+α31(3r + e)

10ϵξ2 1113946

1

0w

5dx1113888 1113889

13

+α31ξϵ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

14 Complexity

α32ξ 11139461

0vwthdxle

α32s2ϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0v5dx1113888 1113889

13

+2α32e5ϵ

ξ2 11139461

0u5dx +

α32(r + 2e)

5ϵξ2 1113946

1

0v5dx

+α32(e +(3r2))

5ϵξ2 1113946

1

0w

5dx +α32ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

2α33ξ 11139461

0wwthdxle

α33sϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0w

5dx1113888 1113889

13

+2α33e5ϵ

ξ2

+α33(5r + 3e)

5ϵξ2 1113946

1

0w


1

0ww

2tdx

α12ξ 11139461

0uvtfdxle

α12ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+α125ϵ

2a +32

1113874 1113875ξ2 11139461

0u5dx

+aα1210ϵ

ξ2 11139461

0v5dx +

α125ϵ

ξ2 11139461

0w

5dx +α12ϵξ2

dminus 11 + ξ1113872 1113873 1113946

1

0uv

2tdx +

α12ϵξ2

dminus 11 + aξ1113872 1113873 1113946

1

0vv

2tdx

α12ξ 11139461

0uwtfdxle

α12ξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+ 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+3α12(a + 1)

10ϵξ2 1113946

1

0u5dx +

aα125ϵ

ξ2 11139461

0v5dx +

α125ϵ

ξ2 11139461

0w


aξ2

+ξ2

1113888 1113889 11139461

0uw

2tdx

α21ξ 11139461

0vutkdxle

α21ξ2ϵ

M231 d

minus 11 d

minus (43)2 b 1113946

1

0u5dx1113888 1113889

13

+ g 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+α215ϵ

ξ2 11139461

0u5dx +

3α2110ϵ

ξ2 11139461

0v5dx +

α21ϵξ2

bdminus 11 + gd

minus 11 + ξ1113872 1113873 1113946

1

0vu

2tdx

α23ξ 11139461

0vwtkdxle

α23ξ2ϵ

M231 d

minus 11 d

minus (43)2 b 1113946

1

0u5dx1113888 1113889

13

+ g 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+α235ϵ

ξ2 11139461

0u5dx +

3α2310ϵ

ξ2 11139461

0v5dx +

α23ϵξ2

bdminus 11 + gd

minus 11 + ξ1113872 1113873 1113946

1

0vw

2tdx

α31ξ 11139461

0wuthdxle

α31sξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α31e5ϵ

ξ2 11139461

0u5dx

+α31ξ

2

2ϵr +

35

e1113874 1113875 11139461

0w

5dx +α31ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0wu

2tdx

α32ξ 11139461

0wvthdxle

α32sξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α32e5ϵ

ξ2 11139461

0u5dx

+α32ξ

2

2ϵr +

35

e1113874 1113875 11139461

0w

5dx +α32ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0wv

2tdx

(75)

Complexity 15

erefore

11139461

01 + p11ξ( 1113857utfdx + ξ 1113946

1

01 + p22( 1113857vtkdx + 1113946

1

0η + p33ξ( 1113857wthdx

+ ξ 11139461

0p12vtfdx + ξ 1113946

1

0p13wtfdx + ξ 1113946

1

0p21utkdx

+ ξ 11139461

0p23wtkdx + ξ 1113946

1

0p31uthdx + ξ 1113946

1

0p32vthdx

le λϵξ 11139461

0(u + v + w) u

2t + v

2t + w

2t1113872 1113873dx +

C8

ϵM0d

minus 11 d

minus 22 (1 + ξ + η)

+C9

ϵM

231 d

minus (43)2 ξ 1 + d

minus 11 + ξ + η1113872 1113873 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx1113890 1113891

13

+C10

ϵξ2 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx

(76)



12

11139461

0P2xxdx minus

ξ2

11139461

0Q

2xxdx minus

η2

11139461

0R2xxdx

+ B1K2dminus 11 d

minus 12 + B2K3d

minus (43)1 z

13


23+ B4K5z

(77)

where z 111393810(u5 + v5 + w5)dx K2 (1 + ξ + η)


2 ) K3 M231 ξ(1 + dminus 1

1 + ξ + η)



zleB5ξminus (52)

11139461

0P52

+ Q52

+ R52

1113872 1113873dx

leB6ξminus (52)

K321 d

minus (32)1 y

12+ B6ξ

minus (52)K

521 d

minus (52)1

z13 leB7ξ

minus (56)K

121 d

minus (12)1 y

16+ B7ξ

minus (56)K

561 d

minus (56)1

z23 leB8ξ

minus (53)K1d

minus 11 y

13+ B8ξ

minus (53)K

531 d

minus (53)1

(78)


431 d

minus (43)1 (|Pxx|22 + K2

1dminus 21 ) So

minusξ2

1113946

1

0

P2xxdx minus

12

1113946

1

0

Q2xxdx minus

η2

1113946

1

0

R2xxdx


431 y

53+(1 + ξ + η)K

21d

minus 21

(79)


2 we have

12


432 y

53

+ A2ξminus (56)

K121 K3d

minus (16)2 y

16

+ A3ξminus (53)

K1K4dminus (13)2 y

13

+ A4ξminus (52)

K321 K5d

minus (12)2 y

12

+ A5 K21(1 + ξ + η) + K2ξ1113960

+ K561 K3ξ

minus (56)d

minus (16)2 + K

531 K4ξ

minus (53)d

minus (12)2

+ K521 K5ξ

minus (52)d

minus (12)2 1113961

(80)



11139461

0d2Px( 1113857

2dx 11139461

0d2Qx( 1113857

2dx 11139461

0d2Rx( 1113857

2dxle 1113958M2 tge 1113957τ2

(81)




16 Complexity

ux

vx

wx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ J

minus 1

Px

Qx

Rx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

J

Pu Pv Pw

Qu Qv Qw

Ru Rv Rw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(82)


d2ux

11138681113868111386811138681113868111386811138681113868 + d2vx

11138681113868111386811138681113868111386811138681113868 + d2wx

11138681113868111386811138681113868111386811138681113868leL d2Px

11138681113868111386811138681113868111386811138681113868 + d2Qx

11138681113868111386811138681113868111386811138681113868 + d2Rx

111386811138681113868111386811138681113868111386811138681113872 1113873

0lt xlt 1 tgt 0

(83)



11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w





11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w














2d1 + 2α11M + α12M + α13M( 1113857

+ λ1113957vM2 α131113957u + ρα31 1113957w( 1113857

2d2 + α21M + 2α22M + α23M( 1113857

+ ρ1113957wM2 α121113957u + λα211113957v( 1113857

2d3 + α31M + α32M + 2α33M( 1113857

(87)



Define

V(u v w) 11139461


u


v

1113957v1113874 11138751113882


1113957w1113874 11138751113883dx

(88)


dV(u v w)

dt minus 1113946

1

0

1113957u

u2 d1 + p11( 1113857u2x +

λ1113957v

v2d2 + p22( 1113857v

2x1113896

+ρ1113957w

w2 d3 + p33( 1113857w2x +

α121113957u

u+λα211113957v

v1113888 1113889uxvx

+α131113957u

u+ρα31 1113957w

w1113874 1113875uxwx

+λα231113957v

v+ρα32 1113957w

w1113888 1113889vxwx1113897dx

minus 11139461

0

v

1113957uu(u minus 1113957u)

2+

bλu

1113957vv(v minus 1113957v)

21113896

+ ρr(w minus 1113957w)2

minusgλ + 1



(89)

Complexity 17


4λρ1113957u1113957v1113957w d1 + p11( 1113857 d2 + p22( 1113857 d3 + p33( 1113857


gt ρ1113957w α121113957uv + λα211113957vu( 11138572

d3 + p33( 1113857

+ λ1113957v α131113957uw + ρα31 1113957wu( 11138572

d2 + p22( 1113857

+ 1113957u λα231113957vw + ρα32 1113957wv( 11138572

d1 + p11( 1113857

(90)



1

0(u minus 1113957u)

2+(v minus 1113957v)

2+(w minus 1113957w)

21113960 1113961dx

dH(u v w)

dtlt 0 (u v w)neE3

(91)


10[(u minus 1113957u)2 + (v minus 1113957v)2 +





t⟶infin

(92)





t⟶infin

(93)



α11 2

α12 1

α13 10

α21 32

α22 12

α23 1

α31 10

α32 2

α33 3

a 12

b 12

g 1

s 3

e 1

r 5

d1 d2 d3≫ 1

(94)


Data Availability




18 Complexity

Acknowledgments


References



























Complexity 19


limt⟶infin

1113946Ω

|nablau|2

+|nablav|2

+|nablaw|2

1113872 1113873dx 0 (23)

limt⟶infin

1113946Ω

(u minus u)2dx 0

limt⟶infin

1113946Ω

(v minus v)2dx 0

(24)


limt⟶infin

1113946Ω

(u minus 1113954u)2dx 0

limt⟶infin

1113946Ω

(v minus 1113954v)2dx 0

limt⟶infin

1113946Ω

(w minus 1113954w)2dx 0

(25)



limt⟶infin

1113954u(t) u

limt⟶infin

1113954v(t) v(26)


1113954uprime tm( 1113857 1113946Ω



(27)


limtm⟶infin

1113954w(t) w (28)





m⟶infin

(29)



C2(Ω)⟶ 0


C2(Ω)⟶ 0


C2(Ω)⟶ 0

m⟶infin

(30)


(2) Define

V(u v w) 1113946Ω


u21113888 11138891113890


v21113888 1113889 +

1e

w1113891dx

(31)


dV(u v w)

dt minus 1113946Ω

d1u2

u2 |nablau|2

+λd2v2

v2|nablav|

21113888 1113889dx minus 1113946

Ω


v


21113896

+(1 minus ag)2 b


u


2+

r

ew

2

+1

b minus g

v

u

1113970


u

v

1113970

v minus v2( 11138571113888 1113889

2⎫⎬

⎭dx

(32)




6 Complexity



zuzt


zuz]

0 x isin zΩ tgt 0


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(33)


P

p11 p12 p13

p21 p22 p23

p31 p33 p33

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (34)

with

p11 2α11us + α12vs + α13ws

p12 α12us

p13 α13us

p21 α21vs

p22 α21us + 2α22vs + α23ws

p23 α23vs

p31 α31ws

p32 α32ws

p33 α31us + α32vs + 2α33ws

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(35)

We denote

L ≔ minus μi(D + P) + J (36)


ϕ λ3 + c1λ2

+ c2λ + c3 (37)

where




2i + β6μi + β7

(38)

with



β3 M22 + M33 + j22j33








(39)








Corollary 1



Complexity 7




wα23 + d2)


eg +


1113969







1113969







β1α13

21113957w3α23α33 + 21113957uα21α33 + 41113957vα22α33 + 2d3α33 + d3α23( 1113857 1113957w

2



β2α13




β3α13

(s minus 2e1113957u)(g + 1113957u)

(40)

Note that

limα13⟶infin

C α13 ]i( 1113857

α13 μi C1μ

2i + C2μi + C31113872 1113873 (41)


eg +


1113969


limα13⟶infin


C2 minus 4C1C3

1113968

2C1lt 0

limα13⟶infin

1113957μ2 0

limα13⟶infin

1113957μ3 minus C2 +

C2 minus 4C1C3

1113968

2C1gt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(42)





⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(43)







(1) Let

8 Complexity

α11 α12 α21 α22 α23 α32 1

α13 12

α31 3

α33 4

a 05

b 1

g 15

s 3

e 2

r 4

d1 3

d2 3

d3 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(44)


0326 0388) c1 5809gt 0 c2 10163gt 0 c3


(2) Let

α11 α12 α21 α22 α23 α31 α32 α33 1

d1 3

d2 3

d3 2

a 05

b 13

g 15

s 3

e 5

r 4

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(45)



(3) Let αij 01 i j 1 2 3 a 28 b 1 g 3 s










(46)






|u|2 leC ux

11138681113868111386811138681113868111386811138681113868132 |u|

231 + |u|11113874 1113875 forallu isinW

12(0 1) (47)

|u|4 leC ux

11138681113868111386811138681113868111386811138681113868122 |u|

121 + |u|11113874 1113875 forallu isinW

12(0 1) (48)

|u|52 leC ux

11138681113868111386811138681113868111386811138681113868252 |u|

351 + |u|11113874 1113875 forallu isinW

12(0 1) (49)

ux

111386811138681113868111386811138681113868111386811138682leC uxx

11138681113868111386811138681113868111386811138681113868352 |u|

251 + |u|11113874 1113875 forallu isinW

22(0 1) (50)



(H1) ablt 1 bltg




(51)


(52)


Complexity 9








ddt

11139461

0u + ρv +

1e

w1113874 1113875dxleC1 minus l 11139461

0u + ρv +

1e

w1113874 1113875dx

(53)


11139461

0udx 1113946

1

0vdx 1113946

1



11139461

0udx 1113946

1

0vdx 1113946

1



12ddt

11139461


1


1

0p11u

2xdx minus 1113946

1


1

0p13uxwxdx + 1113946

1

0uv + au

2v + u

2w1113872 1113873dx

12ddt

11139461


1


1

0p22v

2xdx minus 1113946

1


1

0p23vxwxdx + b 1113946

1

0uvdx

12ddt

11139461

0w


0w

2xdx minus 1113946

1

0p33w

2xdx minus 1113946

1


1

0p32vxwxdx + s 1113946

1

0w

2dx

(56)


12ddt

11139461

0u2

+ v2

+ w2


11139461

0u2x + v

2x + w

2x1113872 1113873dx minus 1113946

1


+(b + 1) 11139461

0uvdx + a 1113946

1

0u2vdx + s 1113946

1

0w

2dx

(57)

where

q ux vx wx( 1113857 p11u2x + p22v

2x + p33w

2x + p12 + p21( 1113857uxvx

+ p23 + p32( 1113857vxwx + p13 + p31( 1113857uxwx

(58)


12ddt

11139461

0u2

+ v2

+ w2


11139461

0u2x + v

2x + w

2x1113872 1113873dx

+(b + 1) 11139461

0uvdx

+ a 11139461

0u2vdx + s 1113946

1

0w

2dx

(59)

10 Complexity



|u|41)leCM20(|ux|22 + M2


a 11139461

0u2vdxle a 1113946

1

0

u4

2εdx + a 1113946

1

0

ε2v2dx

a2CM2

02dlowast

11139461

0v2dx +

dlowast

211139461

0u2xdx +

dlowastM20

2

(60)


11139461

0u2

+ v2

+ w2


211139461

0u2x + v

2x + w

2x1113872 1113873dx

+ C2 11139461

0u2

+ v2

+ w2

1113872 1113873dx +dlowastM2

02

(61)


|u|61 leCM40(|ux|22 + M2

0) erefore

minus 11139461

0u2x + v

2x + w

2x1113872 1113873dxle 3M

20 minus

19CM2

011139461

0u2

+ v2

+ w2

1113872 1113873dx1113890 1113891

3

(62)


12ddt

11139461

0u2

+ v2

+ w2


11139461

0u2

+ v2

+ w2

1113872 1113873dx1113890 1113891

3

+ C2 11139461

0u2

+ v2

+ w2

1113872 1113873dx + 2dlowastM

20

(63)


such that

11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w





11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w



1113957u u

d2

1113957v v

d2

1113957w w

d2

1113957t d1t

(66)






u(x 0) u0(x)

v(x 0) v0(x)

w(x 0) w0(x)

0ltxlt 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(67)




1u minus gdminus 1




11139461

0udx 1113946

1

0vdx 1113946

1

0wdxleM0d

minus 12

11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w

2dxleM1dminus 22


(68)


1


1

0v2tdx

minus η11139461

0w


1


+ 11139461


+ ξ11139461

0p21utk + 1+ p22( 1113857vtk + p23wtk1113858 1113859dx

+ 11139461


(69)

Complexity 11

where y 111393810(P2

x + Q2x + R2


1 2 3) such that


2t + w

2t1113872 1113873 (70)

erefore


1


1


1

0w


1

0(u + v + w) u

2t + v

2t + w

2t1113872 1113873dx

+ 11139461

01 + p11ξ( 1113857utfdx + 1113946

1

0ξ 1 + p22( 1113857vtkdx + 1113946

1

0η + p33ξ( 1113857wthdx

+ 11139461

0p12ξvtfdx + 1113946

1

0p13ξwtfdx + 1113946

1

0p21ξutkdx

+ 11139461

0p23ξwtkdx + 1113946

1

0p31ξuthdx + 1113946

1

0p32ξvthdx

(71)


11139461

0u4dxleM

131 d

minus (23)1 1113946

1

0u5dx1113888 1113889

23

11139461

0u2v2dxleM

131 d

minus (23)1 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

11139461

0u3dxleM

231 d

minus (43)1 1113946

1

0u5dx1113888 1113889

13

11139461

0uv

2dxleM231 d

minus (43)1 1113946

1

0v5dx1113888 1113889

13

11139461

0u3wdxleM

131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

12

11139461

0w

5dx1113888 1113889

16

11139461

0uvwdxleM

231 d

minus (43)2 1113946

1

0u5dx1113888 1113889

16

11139461

0v5dx1113888 1113889

16

11139461

0u2vwdxleM

131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

16

(72)



12 Complexity

minus 11139461

0u2tdxle minus

12

11139461

0P2xxdx + 1113946

1

0f2dx

minus ξ 11139461

0v2tdxle minus

ξ2

11139461

0Q

2xxdx + ξ 1113946

1

0k2dx

minus η11139461

0w

2tdxle minus

η2

11139461

0R2xxdx + η1113946

1

0h2dx

11139461

0f2dxle 2M1d

minus 21 d

minus 22 + a

2ξ2M131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

+ ξ2M131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0w

5dx1113888 1113889

13

+ 2aξdminus 11 d

minus (43)2 M

231 1113946

1

0v5dx1113888 1113889

13

+ 2ξdminus 11 d

minus (23)2 M

231 1113946

1

0u5dx1113888 1113889

16

11139461

0v5dx1113888 1113889

16


131 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

16

ξ 11139461

0k2dxle ξd

minus 21 d

minus 22 M1 b

2+ g

21113872 1113873 + ξ3dminus (23)

2 M131 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

+ ξ2gdminus 11 d

minus (43)2 M

231 1113946

1

0v5dx1113888 1113889

13

η11139461

0h2dxle ηs

2d

minus 21 d

minus 22 M1 + ηd

2ξ2dminus (23)2 M

131 1113946

1

0u5dx1113888 1113889

13

ηr2ξ2dminus (23)

2 M131


2 M131 1113946

1

0u5dx1113888 1113889

23


131 1113946

1

0u5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

12

(73)

us

minus 11139461


1


1

0w

2tdxle minus

12

11139461

0P2xxdx minus

ξ2

11139461

0Q

2xxdx minus

η2

11139461

0R2xxdx + C5(1 + ξ + η)d

minus 21 d

minus 22 M1

+ C6dminus (43)2 M

231 ξ(1 + η) 1113946

1

0u5

+ v5

1113872 1113873dx1113888 1113889

13

+ C7dminus (23)2 M

131 ξ2(1 + η) 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx1113888 1113889

23

(74)

Complexity 13

11139461

0utfdxle d

minus 11 1113946

1

0uutdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ d

minus 11 1113946

1

0utvdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ aξ 1113946

1

0uutvdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ ξ 1113946

1

0uutwdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le dminus 11

12ϵ

11139461

0udx +ϵ2

11139461

0uu

2tdx1113888 1113889 + d

minus 11

12ϵ

11139461

0vdx +ϵ2

11139461

0vu

2tdx1113888 1113889

+ aξ12ϵ

11139461

0uv

2dx +ϵ2

11139461

0uu

2tdx1113888 1113889 + ξ

12ϵ

11139461

0uw

2dx +ϵ2

11139461

0uu

2tdx1113888 1113889


2ϵ

M0 +a

2ϵM

231 d

minus (43)2 ξ 1113946

1

0v5dx1113888 1113889

13

+ξ2ϵ

M231 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+ϵdminus 1

12

11139461

0vu

2tdx +ϵ2

dminus 11 + aξ + ξ1113872 1113873 1113946

1

0uu

2tdx

2α11ξ 11139461

0uutfdxle

α11ϵ

M231 d

minus 11 d

232 1113946

1

0u5dx1113888 1113889

13

+ 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦ +

3α11(a + 1)

5ϵξ2 1113946

1

0u5dx

+2α11a5ϵ

ξ2 11139461

0v5dx +

2α115ϵ

ξ2 11139461

0w

5dx + α11ξϵ 2dminus 11 + aξ + ξ1113872 1113873 1113946

1

0uu

2tdx

2α12ξ 11139461

0vutfdxle

α12ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0v5dx1113888 1113889

13

+α12(a + 1)

5ϵξ2 1113946

1

0u5dx +

2α125ϵ

ξ2 11139461

0w

5dx +α12(3a + 4)

10ϵξ2 1113946

1

0v5dx

α13ξ 11139461

0wutfdxle

α13ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α13(a +(12))

5ϵξ2 1113946

1

0u5dx +

2aα135ϵ

ξ2 11139461

0v5dx

+2α13(a + 1)

5ϵξ2 1113946

1

0w

5dx +α13dminus 1

12ϵξ 1113946

1

0vu

2tdx +ϵ2


2+ 11113872 1113873 1113946

1

0uu

2tdx

ξ 11139461

0vtkdxle

b + g

2ϵM0d

minus 11 d

minus 12 ξ +

ξ2

2ϵM

231 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+bdminus 1

12ϵξ 1113946

1

0uv

2tdx +ϵ2ξ gd

minus 11 + ξ1113872 1113873 1113946

1

0vv

2tdx

α21ξ 11139461

0uvtkdxle

α21(b + g)

2ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+2α215ϵ

ξ2 11139461

0u5dx

+α2110ϵ

ξ2 11139461

0v5dx +

α21bdminus 11

2ϵξ 1113946

1

0uv

2tdx +

α21ϵξ2

gdminus 11 + ξ1113872 1113873 1113946

1

0vv

2tdx

2α22ξ 11139461

0vvtk dx le

α22(b + g)

ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+α225ϵ

ξ2 11139461

0u5dx

+4α225ϵ

ξ2 11139461

0v5dx + α22(b + g)d

minus 11 ϵξ 1113946

1

0vv

2tdx + α22ϵξ

211139461

0uv

2tdx

α23ξ 11139461

0wvtkdxle

α23(b + g)

2ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

minus (43)

+α23ϵξ2

bdminus 11 + ξ1113872 1113873 1113946

1

0uv

2tdx

+α235ϵ

ξ2 11139461

0u5dx + 21113946

1

0v5dx + 21113946

1

0w

5dx1113888 1113889 +α23gdminus 1

12ϵξ 1113946

1

0vv

2tdx

η11139461

0wthdxle

s

ϵd

minus 11 d

minus 12 M0 +

r

2ϵM

231 d

minus (43)2 ηξ 1113946

1

0w

5dx1113888 1113889

13

+e

2ϵM

231 d

minus (43)2 ηξ 1113946

1

0w

5dx1113888 1113889

13ηϵ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

α31ξ 11139461

0uwthdxle

α31s2ϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0u5dx1113888 1113889

13

+α31(r + 2e)

5ϵξ2 1113946

1

0u5dx1113888 1113889

13

+α31(3r + e)

10ϵξ2 1113946

1

0w

5dx1113888 1113889

13

+α31ξϵ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

14 Complexity

α32ξ 11139461

0vwthdxle

α32s2ϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0v5dx1113888 1113889

13

+2α32e5ϵ

ξ2 11139461

0u5dx +

α32(r + 2e)

5ϵξ2 1113946

1

0v5dx

+α32(e +(3r2))

5ϵξ2 1113946

1

0w

5dx +α32ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

2α33ξ 11139461

0wwthdxle

α33sϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0w

5dx1113888 1113889

13

+2α33e5ϵ

ξ2

+α33(5r + 3e)

5ϵξ2 1113946

1

0w


1

0ww

2tdx

α12ξ 11139461

0uvtfdxle

α12ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+α125ϵ

2a +32

1113874 1113875ξ2 11139461

0u5dx

+aα1210ϵ

ξ2 11139461

0v5dx +

α125ϵ

ξ2 11139461

0w

5dx +α12ϵξ2

dminus 11 + ξ1113872 1113873 1113946

1

0uv

2tdx +

α12ϵξ2

dminus 11 + aξ1113872 1113873 1113946

1

0vv

2tdx

α12ξ 11139461

0uwtfdxle

α12ξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+ 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+3α12(a + 1)

10ϵξ2 1113946

1

0u5dx +

aα125ϵ

ξ2 11139461

0v5dx +

α125ϵ

ξ2 11139461

0w


aξ2

+ξ2

1113888 1113889 11139461

0uw

2tdx

α21ξ 11139461

0vutkdxle

α21ξ2ϵ

M231 d

minus 11 d

minus (43)2 b 1113946

1

0u5dx1113888 1113889

13

+ g 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+α215ϵ

ξ2 11139461

0u5dx +

3α2110ϵ

ξ2 11139461

0v5dx +

α21ϵξ2

bdminus 11 + gd

minus 11 + ξ1113872 1113873 1113946

1

0vu

2tdx

α23ξ 11139461

0vwtkdxle

α23ξ2ϵ

M231 d

minus 11 d

minus (43)2 b 1113946

1

0u5dx1113888 1113889

13

+ g 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+α235ϵ

ξ2 11139461

0u5dx +

3α2310ϵ

ξ2 11139461

0v5dx +

α23ϵξ2

bdminus 11 + gd

minus 11 + ξ1113872 1113873 1113946

1

0vw

2tdx

α31ξ 11139461

0wuthdxle

α31sξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α31e5ϵ

ξ2 11139461

0u5dx

+α31ξ

2

2ϵr +

35

e1113874 1113875 11139461

0w

5dx +α31ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0wu

2tdx

α32ξ 11139461

0wvthdxle

α32sξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α32e5ϵ

ξ2 11139461

0u5dx

+α32ξ

2

2ϵr +

35

e1113874 1113875 11139461

0w

5dx +α32ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0wv

2tdx

(75)

Complexity 15

erefore

11139461

01 + p11ξ( 1113857utfdx + ξ 1113946

1

01 + p22( 1113857vtkdx + 1113946

1

0η + p33ξ( 1113857wthdx

+ ξ 11139461

0p12vtfdx + ξ 1113946

1

0p13wtfdx + ξ 1113946

1

0p21utkdx

+ ξ 11139461

0p23wtkdx + ξ 1113946

1

0p31uthdx + ξ 1113946

1

0p32vthdx

le λϵξ 11139461

0(u + v + w) u

2t + v

2t + w

2t1113872 1113873dx +

C8

ϵM0d

minus 11 d

minus 22 (1 + ξ + η)

+C9

ϵM

231 d

minus (43)2 ξ 1 + d

minus 11 + ξ + η1113872 1113873 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx1113890 1113891

13

+C10

ϵξ2 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx

(76)



12

11139461

0P2xxdx minus

ξ2

11139461

0Q

2xxdx minus

η2

11139461

0R2xxdx

+ B1K2dminus 11 d

minus 12 + B2K3d

minus (43)1 z

13


23+ B4K5z

(77)

where z 111393810(u5 + v5 + w5)dx K2 (1 + ξ + η)


2 ) K3 M231 ξ(1 + dminus 1

1 + ξ + η)



zleB5ξminus (52)

11139461

0P52

+ Q52

+ R52

1113872 1113873dx

leB6ξminus (52)

K321 d

minus (32)1 y

12+ B6ξ

minus (52)K

521 d

minus (52)1

z13 leB7ξ

minus (56)K

121 d

minus (12)1 y

16+ B7ξ

minus (56)K

561 d

minus (56)1

z23 leB8ξ

minus (53)K1d

minus 11 y

13+ B8ξ

minus (53)K

531 d

minus (53)1

(78)


431 d

minus (43)1 (|Pxx|22 + K2

1dminus 21 ) So

minusξ2

1113946

1

0

P2xxdx minus

12

1113946

1

0

Q2xxdx minus

η2

1113946

1

0

R2xxdx


431 y

53+(1 + ξ + η)K

21d

minus 21

(79)


2 we have

12


432 y

53

+ A2ξminus (56)

K121 K3d

minus (16)2 y

16

+ A3ξminus (53)

K1K4dminus (13)2 y

13

+ A4ξminus (52)

K321 K5d

minus (12)2 y

12

+ A5 K21(1 + ξ + η) + K2ξ1113960

+ K561 K3ξ

minus (56)d

minus (16)2 + K

531 K4ξ

minus (53)d

minus (12)2

+ K521 K5ξ

minus (52)d

minus (12)2 1113961

(80)



11139461

0d2Px( 1113857

2dx 11139461

0d2Qx( 1113857

2dx 11139461

0d2Rx( 1113857

2dxle 1113958M2 tge 1113957τ2

(81)




16 Complexity

ux

vx

wx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ J

minus 1

Px

Qx

Rx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

J

Pu Pv Pw

Qu Qv Qw

Ru Rv Rw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(82)


d2ux

11138681113868111386811138681113868111386811138681113868 + d2vx

11138681113868111386811138681113868111386811138681113868 + d2wx

11138681113868111386811138681113868111386811138681113868leL d2Px

11138681113868111386811138681113868111386811138681113868 + d2Qx

11138681113868111386811138681113868111386811138681113868 + d2Rx

111386811138681113868111386811138681113868111386811138681113872 1113873

0lt xlt 1 tgt 0

(83)



11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w





11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w














2d1 + 2α11M + α12M + α13M( 1113857

+ λ1113957vM2 α131113957u + ρα31 1113957w( 1113857

2d2 + α21M + 2α22M + α23M( 1113857

+ ρ1113957wM2 α121113957u + λα211113957v( 1113857

2d3 + α31M + α32M + 2α33M( 1113857

(87)



Define

V(u v w) 11139461


u


v

1113957v1113874 11138751113882


1113957w1113874 11138751113883dx

(88)


dV(u v w)

dt minus 1113946

1

0

1113957u

u2 d1 + p11( 1113857u2x +

λ1113957v

v2d2 + p22( 1113857v

2x1113896

+ρ1113957w

w2 d3 + p33( 1113857w2x +

α121113957u

u+λα211113957v

v1113888 1113889uxvx

+α131113957u

u+ρα31 1113957w

w1113874 1113875uxwx

+λα231113957v

v+ρα32 1113957w

w1113888 1113889vxwx1113897dx

minus 11139461

0

v

1113957uu(u minus 1113957u)

2+

bλu

1113957vv(v minus 1113957v)

21113896

+ ρr(w minus 1113957w)2

minusgλ + 1



(89)

Complexity 17


4λρ1113957u1113957v1113957w d1 + p11( 1113857 d2 + p22( 1113857 d3 + p33( 1113857


gt ρ1113957w α121113957uv + λα211113957vu( 11138572

d3 + p33( 1113857

+ λ1113957v α131113957uw + ρα31 1113957wu( 11138572

d2 + p22( 1113857

+ 1113957u λα231113957vw + ρα32 1113957wv( 11138572

d1 + p11( 1113857

(90)



1

0(u minus 1113957u)

2+(v minus 1113957v)

2+(w minus 1113957w)

21113960 1113961dx

dH(u v w)

dtlt 0 (u v w)neE3

(91)


10[(u minus 1113957u)2 + (v minus 1113957v)2 +





t⟶infin

(92)





t⟶infin

(93)



α11 2

α12 1

α13 10

α21 32

α22 12

α23 1

α31 10

α32 2

α33 3

a 12

b 12

g 1

s 3

e 1

r 5

d1 d2 d3≫ 1

(94)


Data Availability




18 Complexity

Acknowledgments


References



























Complexity 19



zuzt


zuz]

0 x isin zΩ tgt 0


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(33)


P

p11 p12 p13

p21 p22 p23

p31 p33 p33

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (34)

with

p11 2α11us + α12vs + α13ws

p12 α12us

p13 α13us

p21 α21vs

p22 α21us + 2α22vs + α23ws

p23 α23vs

p31 α31ws

p32 α32ws

p33 α31us + α32vs + 2α33ws

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(35)

We denote

L ≔ minus μi(D + P) + J (36)


ϕ λ3 + c1λ2

+ c2λ + c3 (37)

where




2i + β6μi + β7

(38)

with



β3 M22 + M33 + j22j33








(39)








Corollary 1



Complexity 7




wα23 + d2)


eg +


1113969







1113969







β1α13

21113957w3α23α33 + 21113957uα21α33 + 41113957vα22α33 + 2d3α33 + d3α23( 1113857 1113957w

2



β2α13




β3α13

(s minus 2e1113957u)(g + 1113957u)

(40)

Note that

limα13⟶infin

C α13 ]i( 1113857

α13 μi C1μ

2i + C2μi + C31113872 1113873 (41)


eg +


1113969


limα13⟶infin


C2 minus 4C1C3

1113968

2C1lt 0

limα13⟶infin

1113957μ2 0

limα13⟶infin

1113957μ3 minus C2 +

C2 minus 4C1C3

1113968

2C1gt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(42)





⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(43)







(1) Let

8 Complexity

α11 α12 α21 α22 α23 α32 1

α13 12

α31 3

α33 4

a 05

b 1

g 15

s 3

e 2

r 4

d1 3

d2 3

d3 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(44)


0326 0388) c1 5809gt 0 c2 10163gt 0 c3


(2) Let

α11 α12 α21 α22 α23 α31 α32 α33 1

d1 3

d2 3

d3 2

a 05

b 13

g 15

s 3

e 5

r 4

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(45)



(3) Let αij 01 i j 1 2 3 a 28 b 1 g 3 s










(46)






|u|2 leC ux

11138681113868111386811138681113868111386811138681113868132 |u|

231 + |u|11113874 1113875 forallu isinW

12(0 1) (47)

|u|4 leC ux

11138681113868111386811138681113868111386811138681113868122 |u|

121 + |u|11113874 1113875 forallu isinW

12(0 1) (48)

|u|52 leC ux

11138681113868111386811138681113868111386811138681113868252 |u|

351 + |u|11113874 1113875 forallu isinW

12(0 1) (49)

ux

111386811138681113868111386811138681113868111386811138682leC uxx

11138681113868111386811138681113868111386811138681113868352 |u|

251 + |u|11113874 1113875 forallu isinW

22(0 1) (50)



(H1) ablt 1 bltg




(51)


(52)


Complexity 9








ddt

11139461

0u + ρv +

1e

w1113874 1113875dxleC1 minus l 11139461

0u + ρv +

1e

w1113874 1113875dx

(53)


11139461

0udx 1113946

1

0vdx 1113946

1



11139461

0udx 1113946

1

0vdx 1113946

1



12ddt

11139461


1


1

0p11u

2xdx minus 1113946

1


1

0p13uxwxdx + 1113946

1

0uv + au

2v + u

2w1113872 1113873dx

12ddt

11139461


1


1

0p22v

2xdx minus 1113946

1


1

0p23vxwxdx + b 1113946

1

0uvdx

12ddt

11139461

0w


0w

2xdx minus 1113946

1

0p33w

2xdx minus 1113946

1


1

0p32vxwxdx + s 1113946

1

0w

2dx

(56)


12ddt

11139461

0u2

+ v2

+ w2


11139461

0u2x + v

2x + w

2x1113872 1113873dx minus 1113946

1


+(b + 1) 11139461

0uvdx + a 1113946

1

0u2vdx + s 1113946

1

0w

2dx

(57)

where

q ux vx wx( 1113857 p11u2x + p22v

2x + p33w

2x + p12 + p21( 1113857uxvx

+ p23 + p32( 1113857vxwx + p13 + p31( 1113857uxwx

(58)


12ddt

11139461

0u2

+ v2

+ w2


11139461

0u2x + v

2x + w

2x1113872 1113873dx

+(b + 1) 11139461

0uvdx

+ a 11139461

0u2vdx + s 1113946

1

0w

2dx

(59)

10 Complexity



|u|41)leCM20(|ux|22 + M2


a 11139461

0u2vdxle a 1113946

1

0

u4

2εdx + a 1113946

1

0

ε2v2dx

a2CM2

02dlowast

11139461

0v2dx +

dlowast

211139461

0u2xdx +

dlowastM20

2

(60)


11139461

0u2

+ v2

+ w2


211139461

0u2x + v

2x + w

2x1113872 1113873dx

+ C2 11139461

0u2

+ v2

+ w2

1113872 1113873dx +dlowastM2

02

(61)


|u|61 leCM40(|ux|22 + M2

0) erefore

minus 11139461

0u2x + v

2x + w

2x1113872 1113873dxle 3M

20 minus

19CM2

011139461

0u2

+ v2

+ w2

1113872 1113873dx1113890 1113891

3

(62)


12ddt

11139461

0u2

+ v2

+ w2


11139461

0u2

+ v2

+ w2

1113872 1113873dx1113890 1113891

3

+ C2 11139461

0u2

+ v2

+ w2

1113872 1113873dx + 2dlowastM

20

(63)


such that

11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w





11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w



1113957u u

d2

1113957v v

d2

1113957w w

d2

1113957t d1t

(66)






u(x 0) u0(x)

v(x 0) v0(x)

w(x 0) w0(x)

0ltxlt 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(67)




1u minus gdminus 1




11139461

0udx 1113946

1

0vdx 1113946

1

0wdxleM0d

minus 12

11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w

2dxleM1dminus 22


(68)


1


1

0v2tdx

minus η11139461

0w


1


+ 11139461


+ ξ11139461

0p21utk + 1+ p22( 1113857vtk + p23wtk1113858 1113859dx

+ 11139461


(69)

Complexity 11

where y 111393810(P2

x + Q2x + R2


1 2 3) such that


2t + w

2t1113872 1113873 (70)

erefore


1


1


1

0w


1

0(u + v + w) u

2t + v

2t + w

2t1113872 1113873dx

+ 11139461

01 + p11ξ( 1113857utfdx + 1113946

1

0ξ 1 + p22( 1113857vtkdx + 1113946

1

0η + p33ξ( 1113857wthdx

+ 11139461

0p12ξvtfdx + 1113946

1

0p13ξwtfdx + 1113946

1

0p21ξutkdx

+ 11139461

0p23ξwtkdx + 1113946

1

0p31ξuthdx + 1113946

1

0p32ξvthdx

(71)


11139461

0u4dxleM

131 d

minus (23)1 1113946

1

0u5dx1113888 1113889

23

11139461

0u2v2dxleM

131 d

minus (23)1 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

11139461

0u3dxleM

231 d

minus (43)1 1113946

1

0u5dx1113888 1113889

13

11139461

0uv

2dxleM231 d

minus (43)1 1113946

1

0v5dx1113888 1113889

13

11139461

0u3wdxleM

131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

12

11139461

0w

5dx1113888 1113889

16

11139461

0uvwdxleM

231 d

minus (43)2 1113946

1

0u5dx1113888 1113889

16

11139461

0v5dx1113888 1113889

16

11139461

0u2vwdxleM

131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

16

(72)



12 Complexity

minus 11139461

0u2tdxle minus

12

11139461

0P2xxdx + 1113946

1

0f2dx

minus ξ 11139461

0v2tdxle minus

ξ2

11139461

0Q

2xxdx + ξ 1113946

1

0k2dx

minus η11139461

0w

2tdxle minus

η2

11139461

0R2xxdx + η1113946

1

0h2dx

11139461

0f2dxle 2M1d

minus 21 d

minus 22 + a

2ξ2M131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

+ ξ2M131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0w

5dx1113888 1113889

13

+ 2aξdminus 11 d

minus (43)2 M

231 1113946

1

0v5dx1113888 1113889

13

+ 2ξdminus 11 d

minus (23)2 M

231 1113946

1

0u5dx1113888 1113889

16

11139461

0v5dx1113888 1113889

16


131 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

16

ξ 11139461

0k2dxle ξd

minus 21 d

minus 22 M1 b

2+ g

21113872 1113873 + ξ3dminus (23)

2 M131 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

+ ξ2gdminus 11 d

minus (43)2 M

231 1113946

1

0v5dx1113888 1113889

13

η11139461

0h2dxle ηs

2d

minus 21 d

minus 22 M1 + ηd

2ξ2dminus (23)2 M

131 1113946

1

0u5dx1113888 1113889

13

ηr2ξ2dminus (23)

2 M131


2 M131 1113946

1

0u5dx1113888 1113889

23


131 1113946

1

0u5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

12

(73)

us

minus 11139461


1


1

0w

2tdxle minus

12

11139461

0P2xxdx minus

ξ2

11139461

0Q

2xxdx minus

η2

11139461

0R2xxdx + C5(1 + ξ + η)d

minus 21 d

minus 22 M1

+ C6dminus (43)2 M

231 ξ(1 + η) 1113946

1

0u5

+ v5

1113872 1113873dx1113888 1113889

13

+ C7dminus (23)2 M

131 ξ2(1 + η) 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx1113888 1113889

23

(74)

Complexity 13

11139461

0utfdxle d

minus 11 1113946

1

0uutdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ d

minus 11 1113946

1

0utvdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ aξ 1113946

1

0uutvdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ ξ 1113946

1

0uutwdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le dminus 11

12ϵ

11139461

0udx +ϵ2

11139461

0uu

2tdx1113888 1113889 + d

minus 11

12ϵ

11139461

0vdx +ϵ2

11139461

0vu

2tdx1113888 1113889

+ aξ12ϵ

11139461

0uv

2dx +ϵ2

11139461

0uu

2tdx1113888 1113889 + ξ

12ϵ

11139461

0uw

2dx +ϵ2

11139461

0uu

2tdx1113888 1113889


2ϵ

M0 +a

2ϵM

231 d

minus (43)2 ξ 1113946

1

0v5dx1113888 1113889

13

+ξ2ϵ

M231 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+ϵdminus 1

12

11139461

0vu

2tdx +ϵ2

dminus 11 + aξ + ξ1113872 1113873 1113946

1

0uu

2tdx

2α11ξ 11139461

0uutfdxle

α11ϵ

M231 d

minus 11 d

232 1113946

1

0u5dx1113888 1113889

13

+ 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦ +

3α11(a + 1)

5ϵξ2 1113946

1

0u5dx

+2α11a5ϵ

ξ2 11139461

0v5dx +

2α115ϵ

ξ2 11139461

0w

5dx + α11ξϵ 2dminus 11 + aξ + ξ1113872 1113873 1113946

1

0uu

2tdx

2α12ξ 11139461

0vutfdxle

α12ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0v5dx1113888 1113889

13

+α12(a + 1)

5ϵξ2 1113946

1

0u5dx +

2α125ϵ

ξ2 11139461

0w

5dx +α12(3a + 4)

10ϵξ2 1113946

1

0v5dx

α13ξ 11139461

0wutfdxle

α13ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α13(a +(12))

5ϵξ2 1113946

1

0u5dx +

2aα135ϵ

ξ2 11139461

0v5dx

+2α13(a + 1)

5ϵξ2 1113946

1

0w

5dx +α13dminus 1

12ϵξ 1113946

1

0vu

2tdx +ϵ2


2+ 11113872 1113873 1113946

1

0uu

2tdx

ξ 11139461

0vtkdxle

b + g

2ϵM0d

minus 11 d

minus 12 ξ +

ξ2

2ϵM

231 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+bdminus 1

12ϵξ 1113946

1

0uv

2tdx +ϵ2ξ gd

minus 11 + ξ1113872 1113873 1113946

1

0vv

2tdx

α21ξ 11139461

0uvtkdxle

α21(b + g)

2ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+2α215ϵ

ξ2 11139461

0u5dx

+α2110ϵ

ξ2 11139461

0v5dx +

α21bdminus 11

2ϵξ 1113946

1

0uv

2tdx +

α21ϵξ2

gdminus 11 + ξ1113872 1113873 1113946

1

0vv

2tdx

2α22ξ 11139461

0vvtk dx le

α22(b + g)

ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+α225ϵ

ξ2 11139461

0u5dx

+4α225ϵ

ξ2 11139461

0v5dx + α22(b + g)d

minus 11 ϵξ 1113946

1

0vv

2tdx + α22ϵξ

211139461

0uv

2tdx

α23ξ 11139461

0wvtkdxle

α23(b + g)

2ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

minus (43)

+α23ϵξ2

bdminus 11 + ξ1113872 1113873 1113946

1

0uv

2tdx

+α235ϵ

ξ2 11139461

0u5dx + 21113946

1

0v5dx + 21113946

1

0w

5dx1113888 1113889 +α23gdminus 1

12ϵξ 1113946

1

0vv

2tdx

η11139461

0wthdxle

s

ϵd

minus 11 d

minus 12 M0 +

r

2ϵM

231 d

minus (43)2 ηξ 1113946

1

0w

5dx1113888 1113889

13

+e

2ϵM

231 d

minus (43)2 ηξ 1113946

1

0w

5dx1113888 1113889

13ηϵ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

α31ξ 11139461

0uwthdxle

α31s2ϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0u5dx1113888 1113889

13

+α31(r + 2e)

5ϵξ2 1113946

1

0u5dx1113888 1113889

13

+α31(3r + e)

10ϵξ2 1113946

1

0w

5dx1113888 1113889

13

+α31ξϵ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

14 Complexity

α32ξ 11139461

0vwthdxle

α32s2ϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0v5dx1113888 1113889

13

+2α32e5ϵ

ξ2 11139461

0u5dx +

α32(r + 2e)

5ϵξ2 1113946

1

0v5dx

+α32(e +(3r2))

5ϵξ2 1113946

1

0w

5dx +α32ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

2α33ξ 11139461

0wwthdxle

α33sϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0w

5dx1113888 1113889

13

+2α33e5ϵ

ξ2

+α33(5r + 3e)

5ϵξ2 1113946

1

0w


1

0ww

2tdx

α12ξ 11139461

0uvtfdxle

α12ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+α125ϵ

2a +32

1113874 1113875ξ2 11139461

0u5dx

+aα1210ϵ

ξ2 11139461

0v5dx +

α125ϵ

ξ2 11139461

0w

5dx +α12ϵξ2

dminus 11 + ξ1113872 1113873 1113946

1

0uv

2tdx +

α12ϵξ2

dminus 11 + aξ1113872 1113873 1113946

1

0vv

2tdx

α12ξ 11139461

0uwtfdxle

α12ξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+ 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+3α12(a + 1)

10ϵξ2 1113946

1

0u5dx +

aα125ϵ

ξ2 11139461

0v5dx +

α125ϵ

ξ2 11139461

0w


aξ2

+ξ2

1113888 1113889 11139461

0uw

2tdx

α21ξ 11139461

0vutkdxle

α21ξ2ϵ

M231 d

minus 11 d

minus (43)2 b 1113946

1

0u5dx1113888 1113889

13

+ g 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+α215ϵ

ξ2 11139461

0u5dx +

3α2110ϵ

ξ2 11139461

0v5dx +

α21ϵξ2

bdminus 11 + gd

minus 11 + ξ1113872 1113873 1113946

1

0vu

2tdx

α23ξ 11139461

0vwtkdxle

α23ξ2ϵ

M231 d

minus 11 d

minus (43)2 b 1113946

1

0u5dx1113888 1113889

13

+ g 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+α235ϵ

ξ2 11139461

0u5dx +

3α2310ϵ

ξ2 11139461

0v5dx +

α23ϵξ2

bdminus 11 + gd

minus 11 + ξ1113872 1113873 1113946

1

0vw

2tdx

α31ξ 11139461

0wuthdxle

α31sξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α31e5ϵ

ξ2 11139461

0u5dx

+α31ξ

2

2ϵr +

35

e1113874 1113875 11139461

0w

5dx +α31ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0wu

2tdx

α32ξ 11139461

0wvthdxle

α32sξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α32e5ϵ

ξ2 11139461

0u5dx

+α32ξ

2

2ϵr +

35

e1113874 1113875 11139461

0w

5dx +α32ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0wv

2tdx

(75)

Complexity 15

erefore

11139461

01 + p11ξ( 1113857utfdx + ξ 1113946

1

01 + p22( 1113857vtkdx + 1113946

1

0η + p33ξ( 1113857wthdx

+ ξ 11139461

0p12vtfdx + ξ 1113946

1

0p13wtfdx + ξ 1113946

1

0p21utkdx

+ ξ 11139461

0p23wtkdx + ξ 1113946

1

0p31uthdx + ξ 1113946

1

0p32vthdx

le λϵξ 11139461

0(u + v + w) u

2t + v

2t + w

2t1113872 1113873dx +

C8

ϵM0d

minus 11 d

minus 22 (1 + ξ + η)

+C9

ϵM

231 d

minus (43)2 ξ 1 + d

minus 11 + ξ + η1113872 1113873 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx1113890 1113891

13

+C10

ϵξ2 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx

(76)



12

11139461

0P2xxdx minus

ξ2

11139461

0Q

2xxdx minus

η2

11139461

0R2xxdx

+ B1K2dminus 11 d

minus 12 + B2K3d

minus (43)1 z

13


23+ B4K5z

(77)

where z 111393810(u5 + v5 + w5)dx K2 (1 + ξ + η)


2 ) K3 M231 ξ(1 + dminus 1

1 + ξ + η)



zleB5ξminus (52)

11139461

0P52

+ Q52

+ R52

1113872 1113873dx

leB6ξminus (52)

K321 d

minus (32)1 y

12+ B6ξ

minus (52)K

521 d

minus (52)1

z13 leB7ξ

minus (56)K

121 d

minus (12)1 y

16+ B7ξ

minus (56)K

561 d

minus (56)1

z23 leB8ξ

minus (53)K1d

minus 11 y

13+ B8ξ

minus (53)K

531 d

minus (53)1

(78)


431 d

minus (43)1 (|Pxx|22 + K2

1dminus 21 ) So

minusξ2

1113946

1

0

P2xxdx minus

12

1113946

1

0

Q2xxdx minus

η2

1113946

1

0

R2xxdx


431 y

53+(1 + ξ + η)K

21d

minus 21

(79)


2 we have

12


432 y

53

+ A2ξminus (56)

K121 K3d

minus (16)2 y

16

+ A3ξminus (53)

K1K4dminus (13)2 y

13

+ A4ξminus (52)

K321 K5d

minus (12)2 y

12

+ A5 K21(1 + ξ + η) + K2ξ1113960

+ K561 K3ξ

minus (56)d

minus (16)2 + K

531 K4ξ

minus (53)d

minus (12)2

+ K521 K5ξ

minus (52)d

minus (12)2 1113961

(80)



11139461

0d2Px( 1113857

2dx 11139461

0d2Qx( 1113857

2dx 11139461

0d2Rx( 1113857

2dxle 1113958M2 tge 1113957τ2

(81)




16 Complexity

ux

vx

wx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ J

minus 1

Px

Qx

Rx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

J

Pu Pv Pw

Qu Qv Qw

Ru Rv Rw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(82)


d2ux

11138681113868111386811138681113868111386811138681113868 + d2vx

11138681113868111386811138681113868111386811138681113868 + d2wx

11138681113868111386811138681113868111386811138681113868leL d2Px

11138681113868111386811138681113868111386811138681113868 + d2Qx

11138681113868111386811138681113868111386811138681113868 + d2Rx

111386811138681113868111386811138681113868111386811138681113872 1113873

0lt xlt 1 tgt 0

(83)



11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w





11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w














2d1 + 2α11M + α12M + α13M( 1113857

+ λ1113957vM2 α131113957u + ρα31 1113957w( 1113857

2d2 + α21M + 2α22M + α23M( 1113857

+ ρ1113957wM2 α121113957u + λα211113957v( 1113857

2d3 + α31M + α32M + 2α33M( 1113857

(87)



Define

V(u v w) 11139461


u


v

1113957v1113874 11138751113882


1113957w1113874 11138751113883dx

(88)


dV(u v w)

dt minus 1113946

1

0

1113957u

u2 d1 + p11( 1113857u2x +

λ1113957v

v2d2 + p22( 1113857v

2x1113896

+ρ1113957w

w2 d3 + p33( 1113857w2x +

α121113957u

u+λα211113957v

v1113888 1113889uxvx

+α131113957u

u+ρα31 1113957w

w1113874 1113875uxwx

+λα231113957v

v+ρα32 1113957w

w1113888 1113889vxwx1113897dx

minus 11139461

0

v

1113957uu(u minus 1113957u)

2+

bλu

1113957vv(v minus 1113957v)

21113896

+ ρr(w minus 1113957w)2

minusgλ + 1



(89)

Complexity 17


4λρ1113957u1113957v1113957w d1 + p11( 1113857 d2 + p22( 1113857 d3 + p33( 1113857


gt ρ1113957w α121113957uv + λα211113957vu( 11138572

d3 + p33( 1113857

+ λ1113957v α131113957uw + ρα31 1113957wu( 11138572

d2 + p22( 1113857

+ 1113957u λα231113957vw + ρα32 1113957wv( 11138572

d1 + p11( 1113857

(90)



1

0(u minus 1113957u)

2+(v minus 1113957v)

2+(w minus 1113957w)

21113960 1113961dx

dH(u v w)

dtlt 0 (u v w)neE3

(91)


10[(u minus 1113957u)2 + (v minus 1113957v)2 +





t⟶infin

(92)





t⟶infin

(93)



α11 2

α12 1

α13 10

α21 32

α22 12

α23 1

α31 10

α32 2

α33 3

a 12

b 12

g 1

s 3

e 1

r 5

d1 d2 d3≫ 1

(94)


Data Availability




18 Complexity

Acknowledgments


References



























Complexity 19




wα23 + d2)


eg +


1113969







1113969







β1α13

21113957w3α23α33 + 21113957uα21α33 + 41113957vα22α33 + 2d3α33 + d3α23( 1113857 1113957w

2



β2α13




β3α13

(s minus 2e1113957u)(g + 1113957u)

(40)

Note that

limα13⟶infin

C α13 ]i( 1113857

α13 μi C1μ

2i + C2μi + C31113872 1113873 (41)


eg +


1113969


limα13⟶infin


C2 minus 4C1C3

1113968

2C1lt 0

limα13⟶infin

1113957μ2 0

limα13⟶infin

1113957μ3 minus C2 +

C2 minus 4C1C3

1113968

2C1gt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(42)





⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(43)







(1) Let

8 Complexity

α11 α12 α21 α22 α23 α32 1

α13 12

α31 3

α33 4

a 05

b 1

g 15

s 3

e 2

r 4

d1 3

d2 3

d3 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(44)


0326 0388) c1 5809gt 0 c2 10163gt 0 c3


(2) Let

α11 α12 α21 α22 α23 α31 α32 α33 1

d1 3

d2 3

d3 2

a 05

b 13

g 15

s 3

e 5

r 4

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(45)



(3) Let αij 01 i j 1 2 3 a 28 b 1 g 3 s










(46)






|u|2 leC ux

11138681113868111386811138681113868111386811138681113868132 |u|

231 + |u|11113874 1113875 forallu isinW

12(0 1) (47)

|u|4 leC ux

11138681113868111386811138681113868111386811138681113868122 |u|

121 + |u|11113874 1113875 forallu isinW

12(0 1) (48)

|u|52 leC ux

11138681113868111386811138681113868111386811138681113868252 |u|

351 + |u|11113874 1113875 forallu isinW

12(0 1) (49)

ux

111386811138681113868111386811138681113868111386811138682leC uxx

11138681113868111386811138681113868111386811138681113868352 |u|

251 + |u|11113874 1113875 forallu isinW

22(0 1) (50)



(H1) ablt 1 bltg




(51)


(52)


Complexity 9








ddt

11139461

0u + ρv +

1e

w1113874 1113875dxleC1 minus l 11139461

0u + ρv +

1e

w1113874 1113875dx

(53)


11139461

0udx 1113946

1

0vdx 1113946

1



11139461

0udx 1113946

1

0vdx 1113946

1



12ddt

11139461


1


1

0p11u

2xdx minus 1113946

1


1

0p13uxwxdx + 1113946

1

0uv + au

2v + u

2w1113872 1113873dx

12ddt

11139461


1


1

0p22v

2xdx minus 1113946

1


1

0p23vxwxdx + b 1113946

1

0uvdx

12ddt

11139461

0w


0w

2xdx minus 1113946

1

0p33w

2xdx minus 1113946

1


1

0p32vxwxdx + s 1113946

1

0w

2dx

(56)


12ddt

11139461

0u2

+ v2

+ w2


11139461

0u2x + v

2x + w

2x1113872 1113873dx minus 1113946

1


+(b + 1) 11139461

0uvdx + a 1113946

1

0u2vdx + s 1113946

1

0w

2dx

(57)

where

q ux vx wx( 1113857 p11u2x + p22v

2x + p33w

2x + p12 + p21( 1113857uxvx

+ p23 + p32( 1113857vxwx + p13 + p31( 1113857uxwx

(58)


12ddt

11139461

0u2

+ v2

+ w2


11139461

0u2x + v

2x + w

2x1113872 1113873dx

+(b + 1) 11139461

0uvdx

+ a 11139461

0u2vdx + s 1113946

1

0w

2dx

(59)

10 Complexity



|u|41)leCM20(|ux|22 + M2


a 11139461

0u2vdxle a 1113946

1

0

u4

2εdx + a 1113946

1

0

ε2v2dx

a2CM2

02dlowast

11139461

0v2dx +

dlowast

211139461

0u2xdx +

dlowastM20

2

(60)


11139461

0u2

+ v2

+ w2


211139461

0u2x + v

2x + w

2x1113872 1113873dx

+ C2 11139461

0u2

+ v2

+ w2

1113872 1113873dx +dlowastM2

02

(61)


|u|61 leCM40(|ux|22 + M2

0) erefore

minus 11139461

0u2x + v

2x + w

2x1113872 1113873dxle 3M

20 minus

19CM2

011139461

0u2

+ v2

+ w2

1113872 1113873dx1113890 1113891

3

(62)


12ddt

11139461

0u2

+ v2

+ w2


11139461

0u2

+ v2

+ w2

1113872 1113873dx1113890 1113891

3

+ C2 11139461

0u2

+ v2

+ w2

1113872 1113873dx + 2dlowastM

20

(63)


such that

11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w





11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w



1113957u u

d2

1113957v v

d2

1113957w w

d2

1113957t d1t

(66)






u(x 0) u0(x)

v(x 0) v0(x)

w(x 0) w0(x)

0ltxlt 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(67)




1u minus gdminus 1




11139461

0udx 1113946

1

0vdx 1113946

1

0wdxleM0d

minus 12

11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w

2dxleM1dminus 22


(68)


1


1

0v2tdx

minus η11139461

0w


1


+ 11139461


+ ξ11139461

0p21utk + 1+ p22( 1113857vtk + p23wtk1113858 1113859dx

+ 11139461


(69)

Complexity 11

where y 111393810(P2

x + Q2x + R2


1 2 3) such that


2t + w

2t1113872 1113873 (70)

erefore


1


1


1

0w


1

0(u + v + w) u

2t + v

2t + w

2t1113872 1113873dx

+ 11139461

01 + p11ξ( 1113857utfdx + 1113946

1

0ξ 1 + p22( 1113857vtkdx + 1113946

1

0η + p33ξ( 1113857wthdx

+ 11139461

0p12ξvtfdx + 1113946

1

0p13ξwtfdx + 1113946

1

0p21ξutkdx

+ 11139461

0p23ξwtkdx + 1113946

1

0p31ξuthdx + 1113946

1

0p32ξvthdx

(71)


11139461

0u4dxleM

131 d

minus (23)1 1113946

1

0u5dx1113888 1113889

23

11139461

0u2v2dxleM

131 d

minus (23)1 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

11139461

0u3dxleM

231 d

minus (43)1 1113946

1

0u5dx1113888 1113889

13

11139461

0uv

2dxleM231 d

minus (43)1 1113946

1

0v5dx1113888 1113889

13

11139461

0u3wdxleM

131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

12

11139461

0w

5dx1113888 1113889

16

11139461

0uvwdxleM

231 d

minus (43)2 1113946

1

0u5dx1113888 1113889

16

11139461

0v5dx1113888 1113889

16

11139461

0u2vwdxleM

131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

16

(72)



12 Complexity

minus 11139461

0u2tdxle minus

12

11139461

0P2xxdx + 1113946

1

0f2dx

minus ξ 11139461

0v2tdxle minus

ξ2

11139461

0Q

2xxdx + ξ 1113946

1

0k2dx

minus η11139461

0w

2tdxle minus

η2

11139461

0R2xxdx + η1113946

1

0h2dx

11139461

0f2dxle 2M1d

minus 21 d

minus 22 + a

2ξ2M131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

+ ξ2M131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0w

5dx1113888 1113889

13

+ 2aξdminus 11 d

minus (43)2 M

231 1113946

1

0v5dx1113888 1113889

13

+ 2ξdminus 11 d

minus (23)2 M

231 1113946

1

0u5dx1113888 1113889

16

11139461

0v5dx1113888 1113889

16


131 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

16

ξ 11139461

0k2dxle ξd

minus 21 d

minus 22 M1 b

2+ g

21113872 1113873 + ξ3dminus (23)

2 M131 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

+ ξ2gdminus 11 d

minus (43)2 M

231 1113946

1

0v5dx1113888 1113889

13

η11139461

0h2dxle ηs

2d

minus 21 d

minus 22 M1 + ηd

2ξ2dminus (23)2 M

131 1113946

1

0u5dx1113888 1113889

13

ηr2ξ2dminus (23)

2 M131


2 M131 1113946

1

0u5dx1113888 1113889

23


131 1113946

1

0u5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

12

(73)

us

minus 11139461


1


1

0w

2tdxle minus

12

11139461

0P2xxdx minus

ξ2

11139461

0Q

2xxdx minus

η2

11139461

0R2xxdx + C5(1 + ξ + η)d

minus 21 d

minus 22 M1

+ C6dminus (43)2 M

231 ξ(1 + η) 1113946

1

0u5

+ v5

1113872 1113873dx1113888 1113889

13

+ C7dminus (23)2 M

131 ξ2(1 + η) 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx1113888 1113889

23

(74)

Complexity 13

11139461

0utfdxle d

minus 11 1113946

1

0uutdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ d

minus 11 1113946

1

0utvdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ aξ 1113946

1

0uutvdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ ξ 1113946

1

0uutwdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le dminus 11

12ϵ

11139461

0udx +ϵ2

11139461

0uu

2tdx1113888 1113889 + d

minus 11

12ϵ

11139461

0vdx +ϵ2

11139461

0vu

2tdx1113888 1113889

+ aξ12ϵ

11139461

0uv

2dx +ϵ2

11139461

0uu

2tdx1113888 1113889 + ξ

12ϵ

11139461

0uw

2dx +ϵ2

11139461

0uu

2tdx1113888 1113889


2ϵ

M0 +a

2ϵM

231 d

minus (43)2 ξ 1113946

1

0v5dx1113888 1113889

13

+ξ2ϵ

M231 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+ϵdminus 1

12

11139461

0vu

2tdx +ϵ2

dminus 11 + aξ + ξ1113872 1113873 1113946

1

0uu

2tdx

2α11ξ 11139461

0uutfdxle

α11ϵ

M231 d

minus 11 d

232 1113946

1

0u5dx1113888 1113889

13

+ 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦ +

3α11(a + 1)

5ϵξ2 1113946

1

0u5dx

+2α11a5ϵ

ξ2 11139461

0v5dx +

2α115ϵ

ξ2 11139461

0w

5dx + α11ξϵ 2dminus 11 + aξ + ξ1113872 1113873 1113946

1

0uu

2tdx

2α12ξ 11139461

0vutfdxle

α12ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0v5dx1113888 1113889

13

+α12(a + 1)

5ϵξ2 1113946

1

0u5dx +

2α125ϵ

ξ2 11139461

0w

5dx +α12(3a + 4)

10ϵξ2 1113946

1

0v5dx

α13ξ 11139461

0wutfdxle

α13ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α13(a +(12))

5ϵξ2 1113946

1

0u5dx +

2aα135ϵ

ξ2 11139461

0v5dx

+2α13(a + 1)

5ϵξ2 1113946

1

0w

5dx +α13dminus 1

12ϵξ 1113946

1

0vu

2tdx +ϵ2


2+ 11113872 1113873 1113946

1

0uu

2tdx

ξ 11139461

0vtkdxle

b + g

2ϵM0d

minus 11 d

minus 12 ξ +

ξ2

2ϵM

231 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+bdminus 1

12ϵξ 1113946

1

0uv

2tdx +ϵ2ξ gd

minus 11 + ξ1113872 1113873 1113946

1

0vv

2tdx

α21ξ 11139461

0uvtkdxle

α21(b + g)

2ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+2α215ϵ

ξ2 11139461

0u5dx

+α2110ϵ

ξ2 11139461

0v5dx +

α21bdminus 11

2ϵξ 1113946

1

0uv

2tdx +

α21ϵξ2

gdminus 11 + ξ1113872 1113873 1113946

1

0vv

2tdx

2α22ξ 11139461

0vvtk dx le

α22(b + g)

ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+α225ϵ

ξ2 11139461

0u5dx

+4α225ϵ

ξ2 11139461

0v5dx + α22(b + g)d

minus 11 ϵξ 1113946

1

0vv

2tdx + α22ϵξ

211139461

0uv

2tdx

α23ξ 11139461

0wvtkdxle

α23(b + g)

2ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

minus (43)

+α23ϵξ2

bdminus 11 + ξ1113872 1113873 1113946

1

0uv

2tdx

+α235ϵ

ξ2 11139461

0u5dx + 21113946

1

0v5dx + 21113946

1

0w

5dx1113888 1113889 +α23gdminus 1

12ϵξ 1113946

1

0vv

2tdx

η11139461

0wthdxle

s

ϵd

minus 11 d

minus 12 M0 +

r

2ϵM

231 d

minus (43)2 ηξ 1113946

1

0w

5dx1113888 1113889

13

+e

2ϵM

231 d

minus (43)2 ηξ 1113946

1

0w

5dx1113888 1113889

13ηϵ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

α31ξ 11139461

0uwthdxle

α31s2ϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0u5dx1113888 1113889

13

+α31(r + 2e)

5ϵξ2 1113946

1

0u5dx1113888 1113889

13

+α31(3r + e)

10ϵξ2 1113946

1

0w

5dx1113888 1113889

13

+α31ξϵ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

14 Complexity

α32ξ 11139461

0vwthdxle

α32s2ϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0v5dx1113888 1113889

13

+2α32e5ϵ

ξ2 11139461

0u5dx +

α32(r + 2e)

5ϵξ2 1113946

1

0v5dx

+α32(e +(3r2))

5ϵξ2 1113946

1

0w

5dx +α32ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

2α33ξ 11139461

0wwthdxle

α33sϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0w

5dx1113888 1113889

13

+2α33e5ϵ

ξ2

+α33(5r + 3e)

5ϵξ2 1113946

1

0w


1

0ww

2tdx

α12ξ 11139461

0uvtfdxle

α12ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+α125ϵ

2a +32

1113874 1113875ξ2 11139461

0u5dx

+aα1210ϵ

ξ2 11139461

0v5dx +

α125ϵ

ξ2 11139461

0w

5dx +α12ϵξ2

dminus 11 + ξ1113872 1113873 1113946

1

0uv

2tdx +

α12ϵξ2

dminus 11 + aξ1113872 1113873 1113946

1

0vv

2tdx

α12ξ 11139461

0uwtfdxle

α12ξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+ 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+3α12(a + 1)

10ϵξ2 1113946

1

0u5dx +

aα125ϵ

ξ2 11139461

0v5dx +

α125ϵ

ξ2 11139461

0w


aξ2

+ξ2

1113888 1113889 11139461

0uw

2tdx

α21ξ 11139461

0vutkdxle

α21ξ2ϵ

M231 d

minus 11 d

minus (43)2 b 1113946

1

0u5dx1113888 1113889

13

+ g 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+α215ϵ

ξ2 11139461

0u5dx +

3α2110ϵ

ξ2 11139461

0v5dx +

α21ϵξ2

bdminus 11 + gd

minus 11 + ξ1113872 1113873 1113946

1

0vu

2tdx

α23ξ 11139461

0vwtkdxle

α23ξ2ϵ

M231 d

minus 11 d

minus (43)2 b 1113946

1

0u5dx1113888 1113889

13

+ g 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+α235ϵ

ξ2 11139461

0u5dx +

3α2310ϵ

ξ2 11139461

0v5dx +

α23ϵξ2

bdminus 11 + gd

minus 11 + ξ1113872 1113873 1113946

1

0vw

2tdx

α31ξ 11139461

0wuthdxle

α31sξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α31e5ϵ

ξ2 11139461

0u5dx

+α31ξ

2

2ϵr +

35

e1113874 1113875 11139461

0w

5dx +α31ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0wu

2tdx

α32ξ 11139461

0wvthdxle

α32sξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α32e5ϵ

ξ2 11139461

0u5dx

+α32ξ

2

2ϵr +

35

e1113874 1113875 11139461

0w

5dx +α32ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0wv

2tdx

(75)

Complexity 15

erefore

11139461

01 + p11ξ( 1113857utfdx + ξ 1113946

1

01 + p22( 1113857vtkdx + 1113946

1

0η + p33ξ( 1113857wthdx

+ ξ 11139461

0p12vtfdx + ξ 1113946

1

0p13wtfdx + ξ 1113946

1

0p21utkdx

+ ξ 11139461

0p23wtkdx + ξ 1113946

1

0p31uthdx + ξ 1113946

1

0p32vthdx

le λϵξ 11139461

0(u + v + w) u

2t + v

2t + w

2t1113872 1113873dx +

C8

ϵM0d

minus 11 d

minus 22 (1 + ξ + η)

+C9

ϵM

231 d

minus (43)2 ξ 1 + d

minus 11 + ξ + η1113872 1113873 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx1113890 1113891

13

+C10

ϵξ2 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx

(76)



12

11139461

0P2xxdx minus

ξ2

11139461

0Q

2xxdx minus

η2

11139461

0R2xxdx

+ B1K2dminus 11 d

minus 12 + B2K3d

minus (43)1 z

13


23+ B4K5z

(77)

where z 111393810(u5 + v5 + w5)dx K2 (1 + ξ + η)


2 ) K3 M231 ξ(1 + dminus 1

1 + ξ + η)



zleB5ξminus (52)

11139461

0P52

+ Q52

+ R52

1113872 1113873dx

leB6ξminus (52)

K321 d

minus (32)1 y

12+ B6ξ

minus (52)K

521 d

minus (52)1

z13 leB7ξ

minus (56)K

121 d

minus (12)1 y

16+ B7ξ

minus (56)K

561 d

minus (56)1

z23 leB8ξ

minus (53)K1d

minus 11 y

13+ B8ξ

minus (53)K

531 d

minus (53)1

(78)


431 d

minus (43)1 (|Pxx|22 + K2

1dminus 21 ) So

minusξ2

1113946

1

0

P2xxdx minus

12

1113946

1

0

Q2xxdx minus

η2

1113946

1

0

R2xxdx


431 y

53+(1 + ξ + η)K

21d

minus 21

(79)


2 we have

12


432 y

53

+ A2ξminus (56)

K121 K3d

minus (16)2 y

16

+ A3ξminus (53)

K1K4dminus (13)2 y

13

+ A4ξminus (52)

K321 K5d

minus (12)2 y

12

+ A5 K21(1 + ξ + η) + K2ξ1113960

+ K561 K3ξ

minus (56)d

minus (16)2 + K

531 K4ξ

minus (53)d

minus (12)2

+ K521 K5ξ

minus (52)d

minus (12)2 1113961

(80)



11139461

0d2Px( 1113857

2dx 11139461

0d2Qx( 1113857

2dx 11139461

0d2Rx( 1113857

2dxle 1113958M2 tge 1113957τ2

(81)




16 Complexity

ux

vx

wx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ J

minus 1

Px

Qx

Rx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

J

Pu Pv Pw

Qu Qv Qw

Ru Rv Rw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(82)


d2ux

11138681113868111386811138681113868111386811138681113868 + d2vx

11138681113868111386811138681113868111386811138681113868 + d2wx

11138681113868111386811138681113868111386811138681113868leL d2Px

11138681113868111386811138681113868111386811138681113868 + d2Qx

11138681113868111386811138681113868111386811138681113868 + d2Rx

111386811138681113868111386811138681113868111386811138681113872 1113873

0lt xlt 1 tgt 0

(83)



11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w





11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w














2d1 + 2α11M + α12M + α13M( 1113857

+ λ1113957vM2 α131113957u + ρα31 1113957w( 1113857

2d2 + α21M + 2α22M + α23M( 1113857

+ ρ1113957wM2 α121113957u + λα211113957v( 1113857

2d3 + α31M + α32M + 2α33M( 1113857

(87)



Define

V(u v w) 11139461


u


v

1113957v1113874 11138751113882


1113957w1113874 11138751113883dx

(88)


dV(u v w)

dt minus 1113946

1

0

1113957u

u2 d1 + p11( 1113857u2x +

λ1113957v

v2d2 + p22( 1113857v

2x1113896

+ρ1113957w

w2 d3 + p33( 1113857w2x +

α121113957u

u+λα211113957v

v1113888 1113889uxvx

+α131113957u

u+ρα31 1113957w

w1113874 1113875uxwx

+λα231113957v

v+ρα32 1113957w

w1113888 1113889vxwx1113897dx

minus 11139461

0

v

1113957uu(u minus 1113957u)

2+

bλu

1113957vv(v minus 1113957v)

21113896

+ ρr(w minus 1113957w)2

minusgλ + 1



(89)

Complexity 17


4λρ1113957u1113957v1113957w d1 + p11( 1113857 d2 + p22( 1113857 d3 + p33( 1113857


gt ρ1113957w α121113957uv + λα211113957vu( 11138572

d3 + p33( 1113857

+ λ1113957v α131113957uw + ρα31 1113957wu( 11138572

d2 + p22( 1113857

+ 1113957u λα231113957vw + ρα32 1113957wv( 11138572

d1 + p11( 1113857

(90)



1

0(u minus 1113957u)

2+(v minus 1113957v)

2+(w minus 1113957w)

21113960 1113961dx

dH(u v w)

dtlt 0 (u v w)neE3

(91)


10[(u minus 1113957u)2 + (v minus 1113957v)2 +





t⟶infin

(92)





t⟶infin

(93)



α11 2

α12 1

α13 10

α21 32

α22 12

α23 1

α31 10

α32 2

α33 3

a 12

b 12

g 1

s 3

e 1

r 5

d1 d2 d3≫ 1

(94)


Data Availability




18 Complexity

Acknowledgments


References



























Complexity 19

α11 α12 α21 α22 α23 α32 1

α13 12

α31 3

α33 4

a 05

b 1

g 15

s 3

e 2

r 4

d1 3

d2 3

d3 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(44)


0326 0388) c1 5809gt 0 c2 10163gt 0 c3


(2) Let

α11 α12 α21 α22 α23 α31 α32 α33 1

d1 3

d2 3

d3 2

a 05

b 13

g 15

s 3

e 5

r 4

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(45)



(3) Let αij 01 i j 1 2 3 a 28 b 1 g 3 s










(46)






|u|2 leC ux

11138681113868111386811138681113868111386811138681113868132 |u|

231 + |u|11113874 1113875 forallu isinW

12(0 1) (47)

|u|4 leC ux

11138681113868111386811138681113868111386811138681113868122 |u|

121 + |u|11113874 1113875 forallu isinW

12(0 1) (48)

|u|52 leC ux

11138681113868111386811138681113868111386811138681113868252 |u|

351 + |u|11113874 1113875 forallu isinW

12(0 1) (49)

ux

111386811138681113868111386811138681113868111386811138682leC uxx

11138681113868111386811138681113868111386811138681113868352 |u|

251 + |u|11113874 1113875 forallu isinW

22(0 1) (50)



(H1) ablt 1 bltg




(51)


(52)


Complexity 9








ddt

11139461

0u + ρv +

1e

w1113874 1113875dxleC1 minus l 11139461

0u + ρv +

1e

w1113874 1113875dx

(53)


11139461

0udx 1113946

1

0vdx 1113946

1



11139461

0udx 1113946

1

0vdx 1113946

1



12ddt

11139461


1


1

0p11u

2xdx minus 1113946

1


1

0p13uxwxdx + 1113946

1

0uv + au

2v + u

2w1113872 1113873dx

12ddt

11139461


1


1

0p22v

2xdx minus 1113946

1


1

0p23vxwxdx + b 1113946

1

0uvdx

12ddt

11139461

0w


0w

2xdx minus 1113946

1

0p33w

2xdx minus 1113946

1


1

0p32vxwxdx + s 1113946

1

0w

2dx

(56)


12ddt

11139461

0u2

+ v2

+ w2


11139461

0u2x + v

2x + w

2x1113872 1113873dx minus 1113946

1


+(b + 1) 11139461

0uvdx + a 1113946

1

0u2vdx + s 1113946

1

0w

2dx

(57)

where

q ux vx wx( 1113857 p11u2x + p22v

2x + p33w

2x + p12 + p21( 1113857uxvx

+ p23 + p32( 1113857vxwx + p13 + p31( 1113857uxwx

(58)


12ddt

11139461

0u2

+ v2

+ w2


11139461

0u2x + v

2x + w

2x1113872 1113873dx

+(b + 1) 11139461

0uvdx

+ a 11139461

0u2vdx + s 1113946

1

0w

2dx

(59)

10 Complexity



|u|41)leCM20(|ux|22 + M2


a 11139461

0u2vdxle a 1113946

1

0

u4

2εdx + a 1113946

1

0

ε2v2dx

a2CM2

02dlowast

11139461

0v2dx +

dlowast

211139461

0u2xdx +

dlowastM20

2

(60)


11139461

0u2

+ v2

+ w2


211139461

0u2x + v

2x + w

2x1113872 1113873dx

+ C2 11139461

0u2

+ v2

+ w2

1113872 1113873dx +dlowastM2

02

(61)


|u|61 leCM40(|ux|22 + M2

0) erefore

minus 11139461

0u2x + v

2x + w

2x1113872 1113873dxle 3M

20 minus

19CM2

011139461

0u2

+ v2

+ w2

1113872 1113873dx1113890 1113891

3

(62)


12ddt

11139461

0u2

+ v2

+ w2


11139461

0u2

+ v2

+ w2

1113872 1113873dx1113890 1113891

3

+ C2 11139461

0u2

+ v2

+ w2

1113872 1113873dx + 2dlowastM

20

(63)


such that

11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w





11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w



1113957u u

d2

1113957v v

d2

1113957w w

d2

1113957t d1t

(66)






u(x 0) u0(x)

v(x 0) v0(x)

w(x 0) w0(x)

0ltxlt 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(67)




1u minus gdminus 1




11139461

0udx 1113946

1

0vdx 1113946

1

0wdxleM0d

minus 12

11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w

2dxleM1dminus 22


(68)


1


1

0v2tdx

minus η11139461

0w


1


+ 11139461


+ ξ11139461

0p21utk + 1+ p22( 1113857vtk + p23wtk1113858 1113859dx

+ 11139461


(69)

Complexity 11

where y 111393810(P2

x + Q2x + R2


1 2 3) such that


2t + w

2t1113872 1113873 (70)

erefore


1


1


1

0w


1

0(u + v + w) u

2t + v

2t + w

2t1113872 1113873dx

+ 11139461

01 + p11ξ( 1113857utfdx + 1113946

1

0ξ 1 + p22( 1113857vtkdx + 1113946

1

0η + p33ξ( 1113857wthdx

+ 11139461

0p12ξvtfdx + 1113946

1

0p13ξwtfdx + 1113946

1

0p21ξutkdx

+ 11139461

0p23ξwtkdx + 1113946

1

0p31ξuthdx + 1113946

1

0p32ξvthdx

(71)


11139461

0u4dxleM

131 d

minus (23)1 1113946

1

0u5dx1113888 1113889

23

11139461

0u2v2dxleM

131 d

minus (23)1 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

11139461

0u3dxleM

231 d

minus (43)1 1113946

1

0u5dx1113888 1113889

13

11139461

0uv

2dxleM231 d

minus (43)1 1113946

1

0v5dx1113888 1113889

13

11139461

0u3wdxleM

131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

12

11139461

0w

5dx1113888 1113889

16

11139461

0uvwdxleM

231 d

minus (43)2 1113946

1

0u5dx1113888 1113889

16

11139461

0v5dx1113888 1113889

16

11139461

0u2vwdxleM

131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

16

(72)



12 Complexity

minus 11139461

0u2tdxle minus

12

11139461

0P2xxdx + 1113946

1

0f2dx

minus ξ 11139461

0v2tdxle minus

ξ2

11139461

0Q

2xxdx + ξ 1113946

1

0k2dx

minus η11139461

0w

2tdxle minus

η2

11139461

0R2xxdx + η1113946

1

0h2dx

11139461

0f2dxle 2M1d

minus 21 d

minus 22 + a

2ξ2M131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

+ ξ2M131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0w

5dx1113888 1113889

13

+ 2aξdminus 11 d

minus (43)2 M

231 1113946

1

0v5dx1113888 1113889

13

+ 2ξdminus 11 d

minus (23)2 M

231 1113946

1

0u5dx1113888 1113889

16

11139461

0v5dx1113888 1113889

16


131 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

16

ξ 11139461

0k2dxle ξd

minus 21 d

minus 22 M1 b

2+ g

21113872 1113873 + ξ3dminus (23)

2 M131 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

+ ξ2gdminus 11 d

minus (43)2 M

231 1113946

1

0v5dx1113888 1113889

13

η11139461

0h2dxle ηs

2d

minus 21 d

minus 22 M1 + ηd

2ξ2dminus (23)2 M

131 1113946

1

0u5dx1113888 1113889

13

ηr2ξ2dminus (23)

2 M131


2 M131 1113946

1

0u5dx1113888 1113889

23


131 1113946

1

0u5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

12

(73)

us

minus 11139461


1


1

0w

2tdxle minus

12

11139461

0P2xxdx minus

ξ2

11139461

0Q

2xxdx minus

η2

11139461

0R2xxdx + C5(1 + ξ + η)d

minus 21 d

minus 22 M1

+ C6dminus (43)2 M

231 ξ(1 + η) 1113946

1

0u5

+ v5

1113872 1113873dx1113888 1113889

13

+ C7dminus (23)2 M

131 ξ2(1 + η) 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx1113888 1113889

23

(74)

Complexity 13

11139461

0utfdxle d

minus 11 1113946

1

0uutdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ d

minus 11 1113946

1

0utvdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ aξ 1113946

1

0uutvdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ ξ 1113946

1

0uutwdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le dminus 11

12ϵ

11139461

0udx +ϵ2

11139461

0uu

2tdx1113888 1113889 + d

minus 11

12ϵ

11139461

0vdx +ϵ2

11139461

0vu

2tdx1113888 1113889

+ aξ12ϵ

11139461

0uv

2dx +ϵ2

11139461

0uu

2tdx1113888 1113889 + ξ

12ϵ

11139461

0uw

2dx +ϵ2

11139461

0uu

2tdx1113888 1113889


2ϵ

M0 +a

2ϵM

231 d

minus (43)2 ξ 1113946

1

0v5dx1113888 1113889

13

+ξ2ϵ

M231 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+ϵdminus 1

12

11139461

0vu

2tdx +ϵ2

dminus 11 + aξ + ξ1113872 1113873 1113946

1

0uu

2tdx

2α11ξ 11139461

0uutfdxle

α11ϵ

M231 d

minus 11 d

232 1113946

1

0u5dx1113888 1113889

13

+ 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦ +

3α11(a + 1)

5ϵξ2 1113946

1

0u5dx

+2α11a5ϵ

ξ2 11139461

0v5dx +

2α115ϵ

ξ2 11139461

0w

5dx + α11ξϵ 2dminus 11 + aξ + ξ1113872 1113873 1113946

1

0uu

2tdx

2α12ξ 11139461

0vutfdxle

α12ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0v5dx1113888 1113889

13

+α12(a + 1)

5ϵξ2 1113946

1

0u5dx +

2α125ϵ

ξ2 11139461

0w

5dx +α12(3a + 4)

10ϵξ2 1113946

1

0v5dx

α13ξ 11139461

0wutfdxle

α13ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α13(a +(12))

5ϵξ2 1113946

1

0u5dx +

2aα135ϵ

ξ2 11139461

0v5dx

+2α13(a + 1)

5ϵξ2 1113946

1

0w

5dx +α13dminus 1

12ϵξ 1113946

1

0vu

2tdx +ϵ2


2+ 11113872 1113873 1113946

1

0uu

2tdx

ξ 11139461

0vtkdxle

b + g

2ϵM0d

minus 11 d

minus 12 ξ +

ξ2

2ϵM

231 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+bdminus 1

12ϵξ 1113946

1

0uv

2tdx +ϵ2ξ gd

minus 11 + ξ1113872 1113873 1113946

1

0vv

2tdx

α21ξ 11139461

0uvtkdxle

α21(b + g)

2ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+2α215ϵ

ξ2 11139461

0u5dx

+α2110ϵ

ξ2 11139461

0v5dx +

α21bdminus 11

2ϵξ 1113946

1

0uv

2tdx +

α21ϵξ2

gdminus 11 + ξ1113872 1113873 1113946

1

0vv

2tdx

2α22ξ 11139461

0vvtk dx le

α22(b + g)

ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+α225ϵ

ξ2 11139461

0u5dx

+4α225ϵ

ξ2 11139461

0v5dx + α22(b + g)d

minus 11 ϵξ 1113946

1

0vv

2tdx + α22ϵξ

211139461

0uv

2tdx

α23ξ 11139461

0wvtkdxle

α23(b + g)

2ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

minus (43)

+α23ϵξ2

bdminus 11 + ξ1113872 1113873 1113946

1

0uv

2tdx

+α235ϵ

ξ2 11139461

0u5dx + 21113946

1

0v5dx + 21113946

1

0w

5dx1113888 1113889 +α23gdminus 1

12ϵξ 1113946

1

0vv

2tdx

η11139461

0wthdxle

s

ϵd

minus 11 d

minus 12 M0 +

r

2ϵM

231 d

minus (43)2 ηξ 1113946

1

0w

5dx1113888 1113889

13

+e

2ϵM

231 d

minus (43)2 ηξ 1113946

1

0w

5dx1113888 1113889

13ηϵ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

α31ξ 11139461

0uwthdxle

α31s2ϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0u5dx1113888 1113889

13

+α31(r + 2e)

5ϵξ2 1113946

1

0u5dx1113888 1113889

13

+α31(3r + e)

10ϵξ2 1113946

1

0w

5dx1113888 1113889

13

+α31ξϵ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

14 Complexity

α32ξ 11139461

0vwthdxle

α32s2ϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0v5dx1113888 1113889

13

+2α32e5ϵ

ξ2 11139461

0u5dx +

α32(r + 2e)

5ϵξ2 1113946

1

0v5dx

+α32(e +(3r2))

5ϵξ2 1113946

1

0w

5dx +α32ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

2α33ξ 11139461

0wwthdxle

α33sϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0w

5dx1113888 1113889

13

+2α33e5ϵ

ξ2

+α33(5r + 3e)

5ϵξ2 1113946

1

0w


1

0ww

2tdx

α12ξ 11139461

0uvtfdxle

α12ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+α125ϵ

2a +32

1113874 1113875ξ2 11139461

0u5dx

+aα1210ϵ

ξ2 11139461

0v5dx +

α125ϵ

ξ2 11139461

0w

5dx +α12ϵξ2

dminus 11 + ξ1113872 1113873 1113946

1

0uv

2tdx +

α12ϵξ2

dminus 11 + aξ1113872 1113873 1113946

1

0vv

2tdx

α12ξ 11139461

0uwtfdxle

α12ξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+ 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+3α12(a + 1)

10ϵξ2 1113946

1

0u5dx +

aα125ϵ

ξ2 11139461

0v5dx +

α125ϵ

ξ2 11139461

0w


aξ2

+ξ2

1113888 1113889 11139461

0uw

2tdx

α21ξ 11139461

0vutkdxle

α21ξ2ϵ

M231 d

minus 11 d

minus (43)2 b 1113946

1

0u5dx1113888 1113889

13

+ g 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+α215ϵ

ξ2 11139461

0u5dx +

3α2110ϵ

ξ2 11139461

0v5dx +

α21ϵξ2

bdminus 11 + gd

minus 11 + ξ1113872 1113873 1113946

1

0vu

2tdx

α23ξ 11139461

0vwtkdxle

α23ξ2ϵ

M231 d

minus 11 d

minus (43)2 b 1113946

1

0u5dx1113888 1113889

13

+ g 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+α235ϵ

ξ2 11139461

0u5dx +

3α2310ϵ

ξ2 11139461

0v5dx +

α23ϵξ2

bdminus 11 + gd

minus 11 + ξ1113872 1113873 1113946

1

0vw

2tdx

α31ξ 11139461

0wuthdxle

α31sξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α31e5ϵ

ξ2 11139461

0u5dx

+α31ξ

2

2ϵr +

35

e1113874 1113875 11139461

0w

5dx +α31ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0wu

2tdx

α32ξ 11139461

0wvthdxle

α32sξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α32e5ϵ

ξ2 11139461

0u5dx

+α32ξ

2

2ϵr +

35

e1113874 1113875 11139461

0w

5dx +α32ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0wv

2tdx

(75)

Complexity 15

erefore

11139461

01 + p11ξ( 1113857utfdx + ξ 1113946

1

01 + p22( 1113857vtkdx + 1113946

1

0η + p33ξ( 1113857wthdx

+ ξ 11139461

0p12vtfdx + ξ 1113946

1

0p13wtfdx + ξ 1113946

1

0p21utkdx

+ ξ 11139461

0p23wtkdx + ξ 1113946

1

0p31uthdx + ξ 1113946

1

0p32vthdx

le λϵξ 11139461

0(u + v + w) u

2t + v

2t + w

2t1113872 1113873dx +

C8

ϵM0d

minus 11 d

minus 22 (1 + ξ + η)

+C9

ϵM

231 d

minus (43)2 ξ 1 + d

minus 11 + ξ + η1113872 1113873 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx1113890 1113891

13

+C10

ϵξ2 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx

(76)



12

11139461

0P2xxdx minus

ξ2

11139461

0Q

2xxdx minus

η2

11139461

0R2xxdx

+ B1K2dminus 11 d

minus 12 + B2K3d

minus (43)1 z

13


23+ B4K5z

(77)

where z 111393810(u5 + v5 + w5)dx K2 (1 + ξ + η)


2 ) K3 M231 ξ(1 + dminus 1

1 + ξ + η)



zleB5ξminus (52)

11139461

0P52

+ Q52

+ R52

1113872 1113873dx

leB6ξminus (52)

K321 d

minus (32)1 y

12+ B6ξ

minus (52)K

521 d

minus (52)1

z13 leB7ξ

minus (56)K

121 d

minus (12)1 y

16+ B7ξ

minus (56)K

561 d

minus (56)1

z23 leB8ξ

minus (53)K1d

minus 11 y

13+ B8ξ

minus (53)K

531 d

minus (53)1

(78)


431 d

minus (43)1 (|Pxx|22 + K2

1dminus 21 ) So

minusξ2

1113946

1

0

P2xxdx minus

12

1113946

1

0

Q2xxdx minus

η2

1113946

1

0

R2xxdx


431 y

53+(1 + ξ + η)K

21d

minus 21

(79)


2 we have

12


432 y

53

+ A2ξminus (56)

K121 K3d

minus (16)2 y

16

+ A3ξminus (53)

K1K4dminus (13)2 y

13

+ A4ξminus (52)

K321 K5d

minus (12)2 y

12

+ A5 K21(1 + ξ + η) + K2ξ1113960

+ K561 K3ξ

minus (56)d

minus (16)2 + K

531 K4ξ

minus (53)d

minus (12)2

+ K521 K5ξ

minus (52)d

minus (12)2 1113961

(80)



11139461

0d2Px( 1113857

2dx 11139461

0d2Qx( 1113857

2dx 11139461

0d2Rx( 1113857

2dxle 1113958M2 tge 1113957τ2

(81)




16 Complexity

ux

vx

wx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ J

minus 1

Px

Qx

Rx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

J

Pu Pv Pw

Qu Qv Qw

Ru Rv Rw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(82)


d2ux

11138681113868111386811138681113868111386811138681113868 + d2vx

11138681113868111386811138681113868111386811138681113868 + d2wx

11138681113868111386811138681113868111386811138681113868leL d2Px

11138681113868111386811138681113868111386811138681113868 + d2Qx

11138681113868111386811138681113868111386811138681113868 + d2Rx

111386811138681113868111386811138681113868111386811138681113872 1113873

0lt xlt 1 tgt 0

(83)



11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w





11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w














2d1 + 2α11M + α12M + α13M( 1113857

+ λ1113957vM2 α131113957u + ρα31 1113957w( 1113857

2d2 + α21M + 2α22M + α23M( 1113857

+ ρ1113957wM2 α121113957u + λα211113957v( 1113857

2d3 + α31M + α32M + 2α33M( 1113857

(87)



Define

V(u v w) 11139461


u


v

1113957v1113874 11138751113882


1113957w1113874 11138751113883dx

(88)


dV(u v w)

dt minus 1113946

1

0

1113957u

u2 d1 + p11( 1113857u2x +

λ1113957v

v2d2 + p22( 1113857v

2x1113896

+ρ1113957w

w2 d3 + p33( 1113857w2x +

α121113957u

u+λα211113957v

v1113888 1113889uxvx

+α131113957u

u+ρα31 1113957w

w1113874 1113875uxwx

+λα231113957v

v+ρα32 1113957w

w1113888 1113889vxwx1113897dx

minus 11139461

0

v

1113957uu(u minus 1113957u)

2+

bλu

1113957vv(v minus 1113957v)

21113896

+ ρr(w minus 1113957w)2

minusgλ + 1



(89)

Complexity 17


4λρ1113957u1113957v1113957w d1 + p11( 1113857 d2 + p22( 1113857 d3 + p33( 1113857


gt ρ1113957w α121113957uv + λα211113957vu( 11138572

d3 + p33( 1113857

+ λ1113957v α131113957uw + ρα31 1113957wu( 11138572

d2 + p22( 1113857

+ 1113957u λα231113957vw + ρα32 1113957wv( 11138572

d1 + p11( 1113857

(90)



1

0(u minus 1113957u)

2+(v minus 1113957v)

2+(w minus 1113957w)

21113960 1113961dx

dH(u v w)

dtlt 0 (u v w)neE3

(91)


10[(u minus 1113957u)2 + (v minus 1113957v)2 +





t⟶infin

(92)





t⟶infin

(93)



α11 2

α12 1

α13 10

α21 32

α22 12

α23 1

α31 10

α32 2

α33 3

a 12

b 12

g 1

s 3

e 1

r 5

d1 d2 d3≫ 1

(94)


Data Availability




18 Complexity

Acknowledgments


References



























Complexity 19








ddt

11139461

0u + ρv +

1e

w1113874 1113875dxleC1 minus l 11139461

0u + ρv +

1e

w1113874 1113875dx

(53)


11139461

0udx 1113946

1

0vdx 1113946

1



11139461

0udx 1113946

1

0vdx 1113946

1



12ddt

11139461


1


1

0p11u

2xdx minus 1113946

1


1

0p13uxwxdx + 1113946

1

0uv + au

2v + u

2w1113872 1113873dx

12ddt

11139461


1


1

0p22v

2xdx minus 1113946

1


1

0p23vxwxdx + b 1113946

1

0uvdx

12ddt

11139461

0w


0w

2xdx minus 1113946

1

0p33w

2xdx minus 1113946

1


1

0p32vxwxdx + s 1113946

1

0w

2dx

(56)


12ddt

11139461

0u2

+ v2

+ w2


11139461

0u2x + v

2x + w

2x1113872 1113873dx minus 1113946

1


+(b + 1) 11139461

0uvdx + a 1113946

1

0u2vdx + s 1113946

1

0w

2dx

(57)

where

q ux vx wx( 1113857 p11u2x + p22v

2x + p33w

2x + p12 + p21( 1113857uxvx

+ p23 + p32( 1113857vxwx + p13 + p31( 1113857uxwx

(58)


12ddt

11139461

0u2

+ v2

+ w2


11139461

0u2x + v

2x + w

2x1113872 1113873dx

+(b + 1) 11139461

0uvdx

+ a 11139461

0u2vdx + s 1113946

1

0w

2dx

(59)

10 Complexity



|u|41)leCM20(|ux|22 + M2


a 11139461

0u2vdxle a 1113946

1

0

u4

2εdx + a 1113946

1

0

ε2v2dx

a2CM2

02dlowast

11139461

0v2dx +

dlowast

211139461

0u2xdx +

dlowastM20

2

(60)


11139461

0u2

+ v2

+ w2


211139461

0u2x + v

2x + w

2x1113872 1113873dx

+ C2 11139461

0u2

+ v2

+ w2

1113872 1113873dx +dlowastM2

02

(61)


|u|61 leCM40(|ux|22 + M2

0) erefore

minus 11139461

0u2x + v

2x + w

2x1113872 1113873dxle 3M

20 minus

19CM2

011139461

0u2

+ v2

+ w2

1113872 1113873dx1113890 1113891

3

(62)


12ddt

11139461

0u2

+ v2

+ w2


11139461

0u2

+ v2

+ w2

1113872 1113873dx1113890 1113891

3

+ C2 11139461

0u2

+ v2

+ w2

1113872 1113873dx + 2dlowastM

20

(63)


such that

11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w





11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w



1113957u u

d2

1113957v v

d2

1113957w w

d2

1113957t d1t

(66)






u(x 0) u0(x)

v(x 0) v0(x)

w(x 0) w0(x)

0ltxlt 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(67)




1u minus gdminus 1




11139461

0udx 1113946

1

0vdx 1113946

1

0wdxleM0d

minus 12

11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w

2dxleM1dminus 22


(68)


1


1

0v2tdx

minus η11139461

0w


1


+ 11139461


+ ξ11139461

0p21utk + 1+ p22( 1113857vtk + p23wtk1113858 1113859dx

+ 11139461


(69)

Complexity 11

where y 111393810(P2

x + Q2x + R2


1 2 3) such that


2t + w

2t1113872 1113873 (70)

erefore


1


1


1

0w


1

0(u + v + w) u

2t + v

2t + w

2t1113872 1113873dx

+ 11139461

01 + p11ξ( 1113857utfdx + 1113946

1

0ξ 1 + p22( 1113857vtkdx + 1113946

1

0η + p33ξ( 1113857wthdx

+ 11139461

0p12ξvtfdx + 1113946

1

0p13ξwtfdx + 1113946

1

0p21ξutkdx

+ 11139461

0p23ξwtkdx + 1113946

1

0p31ξuthdx + 1113946

1

0p32ξvthdx

(71)


11139461

0u4dxleM

131 d

minus (23)1 1113946

1

0u5dx1113888 1113889

23

11139461

0u2v2dxleM

131 d

minus (23)1 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

11139461

0u3dxleM

231 d

minus (43)1 1113946

1

0u5dx1113888 1113889

13

11139461

0uv

2dxleM231 d

minus (43)1 1113946

1

0v5dx1113888 1113889

13

11139461

0u3wdxleM

131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

12

11139461

0w

5dx1113888 1113889

16

11139461

0uvwdxleM

231 d

minus (43)2 1113946

1

0u5dx1113888 1113889

16

11139461

0v5dx1113888 1113889

16

11139461

0u2vwdxleM

131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

16

(72)



12 Complexity

minus 11139461

0u2tdxle minus

12

11139461

0P2xxdx + 1113946

1

0f2dx

minus ξ 11139461

0v2tdxle minus

ξ2

11139461

0Q

2xxdx + ξ 1113946

1

0k2dx

minus η11139461

0w

2tdxle minus

η2

11139461

0R2xxdx + η1113946

1

0h2dx

11139461

0f2dxle 2M1d

minus 21 d

minus 22 + a

2ξ2M131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

+ ξ2M131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0w

5dx1113888 1113889

13

+ 2aξdminus 11 d

minus (43)2 M

231 1113946

1

0v5dx1113888 1113889

13

+ 2ξdminus 11 d

minus (23)2 M

231 1113946

1

0u5dx1113888 1113889

16

11139461

0v5dx1113888 1113889

16


131 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

16

ξ 11139461

0k2dxle ξd

minus 21 d

minus 22 M1 b

2+ g

21113872 1113873 + ξ3dminus (23)

2 M131 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

+ ξ2gdminus 11 d

minus (43)2 M

231 1113946

1

0v5dx1113888 1113889

13

η11139461

0h2dxle ηs

2d

minus 21 d

minus 22 M1 + ηd

2ξ2dminus (23)2 M

131 1113946

1

0u5dx1113888 1113889

13

ηr2ξ2dminus (23)

2 M131


2 M131 1113946

1

0u5dx1113888 1113889

23


131 1113946

1

0u5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

12

(73)

us

minus 11139461


1


1

0w

2tdxle minus

12

11139461

0P2xxdx minus

ξ2

11139461

0Q

2xxdx minus

η2

11139461

0R2xxdx + C5(1 + ξ + η)d

minus 21 d

minus 22 M1

+ C6dminus (43)2 M

231 ξ(1 + η) 1113946

1

0u5

+ v5

1113872 1113873dx1113888 1113889

13

+ C7dminus (23)2 M

131 ξ2(1 + η) 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx1113888 1113889

23

(74)

Complexity 13

11139461

0utfdxle d

minus 11 1113946

1

0uutdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ d

minus 11 1113946

1

0utvdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ aξ 1113946

1

0uutvdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ ξ 1113946

1

0uutwdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le dminus 11

12ϵ

11139461

0udx +ϵ2

11139461

0uu

2tdx1113888 1113889 + d

minus 11

12ϵ

11139461

0vdx +ϵ2

11139461

0vu

2tdx1113888 1113889

+ aξ12ϵ

11139461

0uv

2dx +ϵ2

11139461

0uu

2tdx1113888 1113889 + ξ

12ϵ

11139461

0uw

2dx +ϵ2

11139461

0uu

2tdx1113888 1113889


2ϵ

M0 +a

2ϵM

231 d

minus (43)2 ξ 1113946

1

0v5dx1113888 1113889

13

+ξ2ϵ

M231 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+ϵdminus 1

12

11139461

0vu

2tdx +ϵ2

dminus 11 + aξ + ξ1113872 1113873 1113946

1

0uu

2tdx

2α11ξ 11139461

0uutfdxle

α11ϵ

M231 d

minus 11 d

232 1113946

1

0u5dx1113888 1113889

13

+ 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦ +

3α11(a + 1)

5ϵξ2 1113946

1

0u5dx

+2α11a5ϵ

ξ2 11139461

0v5dx +

2α115ϵ

ξ2 11139461

0w

5dx + α11ξϵ 2dminus 11 + aξ + ξ1113872 1113873 1113946

1

0uu

2tdx

2α12ξ 11139461

0vutfdxle

α12ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0v5dx1113888 1113889

13

+α12(a + 1)

5ϵξ2 1113946

1

0u5dx +

2α125ϵ

ξ2 11139461

0w

5dx +α12(3a + 4)

10ϵξ2 1113946

1

0v5dx

α13ξ 11139461

0wutfdxle

α13ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α13(a +(12))

5ϵξ2 1113946

1

0u5dx +

2aα135ϵ

ξ2 11139461

0v5dx

+2α13(a + 1)

5ϵξ2 1113946

1

0w

5dx +α13dminus 1

12ϵξ 1113946

1

0vu

2tdx +ϵ2


2+ 11113872 1113873 1113946

1

0uu

2tdx

ξ 11139461

0vtkdxle

b + g

2ϵM0d

minus 11 d

minus 12 ξ +

ξ2

2ϵM

231 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+bdminus 1

12ϵξ 1113946

1

0uv

2tdx +ϵ2ξ gd

minus 11 + ξ1113872 1113873 1113946

1

0vv

2tdx

α21ξ 11139461

0uvtkdxle

α21(b + g)

2ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+2α215ϵ

ξ2 11139461

0u5dx

+α2110ϵ

ξ2 11139461

0v5dx +

α21bdminus 11

2ϵξ 1113946

1

0uv

2tdx +

α21ϵξ2

gdminus 11 + ξ1113872 1113873 1113946

1

0vv

2tdx

2α22ξ 11139461

0vvtk dx le

α22(b + g)

ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+α225ϵ

ξ2 11139461

0u5dx

+4α225ϵ

ξ2 11139461

0v5dx + α22(b + g)d

minus 11 ϵξ 1113946

1

0vv

2tdx + α22ϵξ

211139461

0uv

2tdx

α23ξ 11139461

0wvtkdxle

α23(b + g)

2ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

minus (43)

+α23ϵξ2

bdminus 11 + ξ1113872 1113873 1113946

1

0uv

2tdx

+α235ϵ

ξ2 11139461

0u5dx + 21113946

1

0v5dx + 21113946

1

0w

5dx1113888 1113889 +α23gdminus 1

12ϵξ 1113946

1

0vv

2tdx

η11139461

0wthdxle

s

ϵd

minus 11 d

minus 12 M0 +

r

2ϵM

231 d

minus (43)2 ηξ 1113946

1

0w

5dx1113888 1113889

13

+e

2ϵM

231 d

minus (43)2 ηξ 1113946

1

0w

5dx1113888 1113889

13ηϵ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

α31ξ 11139461

0uwthdxle

α31s2ϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0u5dx1113888 1113889

13

+α31(r + 2e)

5ϵξ2 1113946

1

0u5dx1113888 1113889

13

+α31(3r + e)

10ϵξ2 1113946

1

0w

5dx1113888 1113889

13

+α31ξϵ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

14 Complexity

α32ξ 11139461

0vwthdxle

α32s2ϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0v5dx1113888 1113889

13

+2α32e5ϵ

ξ2 11139461

0u5dx +

α32(r + 2e)

5ϵξ2 1113946

1

0v5dx

+α32(e +(3r2))

5ϵξ2 1113946

1

0w

5dx +α32ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

2α33ξ 11139461

0wwthdxle

α33sϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0w

5dx1113888 1113889

13

+2α33e5ϵ

ξ2

+α33(5r + 3e)

5ϵξ2 1113946

1

0w


1

0ww

2tdx

α12ξ 11139461

0uvtfdxle

α12ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+α125ϵ

2a +32

1113874 1113875ξ2 11139461

0u5dx

+aα1210ϵ

ξ2 11139461

0v5dx +

α125ϵ

ξ2 11139461

0w

5dx +α12ϵξ2

dminus 11 + ξ1113872 1113873 1113946

1

0uv

2tdx +

α12ϵξ2

dminus 11 + aξ1113872 1113873 1113946

1

0vv

2tdx

α12ξ 11139461

0uwtfdxle

α12ξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+ 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+3α12(a + 1)

10ϵξ2 1113946

1

0u5dx +

aα125ϵ

ξ2 11139461

0v5dx +

α125ϵ

ξ2 11139461

0w


aξ2

+ξ2

1113888 1113889 11139461

0uw

2tdx

α21ξ 11139461

0vutkdxle

α21ξ2ϵ

M231 d

minus 11 d

minus (43)2 b 1113946

1

0u5dx1113888 1113889

13

+ g 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+α215ϵ

ξ2 11139461

0u5dx +

3α2110ϵ

ξ2 11139461

0v5dx +

α21ϵξ2

bdminus 11 + gd

minus 11 + ξ1113872 1113873 1113946

1

0vu

2tdx

α23ξ 11139461

0vwtkdxle

α23ξ2ϵ

M231 d

minus 11 d

minus (43)2 b 1113946

1

0u5dx1113888 1113889

13

+ g 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+α235ϵ

ξ2 11139461

0u5dx +

3α2310ϵ

ξ2 11139461

0v5dx +

α23ϵξ2

bdminus 11 + gd

minus 11 + ξ1113872 1113873 1113946

1

0vw

2tdx

α31ξ 11139461

0wuthdxle

α31sξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α31e5ϵ

ξ2 11139461

0u5dx

+α31ξ

2

2ϵr +

35

e1113874 1113875 11139461

0w

5dx +α31ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0wu

2tdx

α32ξ 11139461

0wvthdxle

α32sξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α32e5ϵ

ξ2 11139461

0u5dx

+α32ξ

2

2ϵr +

35

e1113874 1113875 11139461

0w

5dx +α32ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0wv

2tdx

(75)

Complexity 15

erefore

11139461

01 + p11ξ( 1113857utfdx + ξ 1113946

1

01 + p22( 1113857vtkdx + 1113946

1

0η + p33ξ( 1113857wthdx

+ ξ 11139461

0p12vtfdx + ξ 1113946

1

0p13wtfdx + ξ 1113946

1

0p21utkdx

+ ξ 11139461

0p23wtkdx + ξ 1113946

1

0p31uthdx + ξ 1113946

1

0p32vthdx

le λϵξ 11139461

0(u + v + w) u

2t + v

2t + w

2t1113872 1113873dx +

C8

ϵM0d

minus 11 d

minus 22 (1 + ξ + η)

+C9

ϵM

231 d

minus (43)2 ξ 1 + d

minus 11 + ξ + η1113872 1113873 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx1113890 1113891

13

+C10

ϵξ2 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx

(76)



12

11139461

0P2xxdx minus

ξ2

11139461

0Q

2xxdx minus

η2

11139461

0R2xxdx

+ B1K2dminus 11 d

minus 12 + B2K3d

minus (43)1 z

13


23+ B4K5z

(77)

where z 111393810(u5 + v5 + w5)dx K2 (1 + ξ + η)


2 ) K3 M231 ξ(1 + dminus 1

1 + ξ + η)



zleB5ξminus (52)

11139461

0P52

+ Q52

+ R52

1113872 1113873dx

leB6ξminus (52)

K321 d

minus (32)1 y

12+ B6ξ

minus (52)K

521 d

minus (52)1

z13 leB7ξ

minus (56)K

121 d

minus (12)1 y

16+ B7ξ

minus (56)K

561 d

minus (56)1

z23 leB8ξ

minus (53)K1d

minus 11 y

13+ B8ξ

minus (53)K

531 d

minus (53)1

(78)


431 d

minus (43)1 (|Pxx|22 + K2

1dminus 21 ) So

minusξ2

1113946

1

0

P2xxdx minus

12

1113946

1

0

Q2xxdx minus

η2

1113946

1

0

R2xxdx


431 y

53+(1 + ξ + η)K

21d

minus 21

(79)


2 we have

12


432 y

53

+ A2ξminus (56)

K121 K3d

minus (16)2 y

16

+ A3ξminus (53)

K1K4dminus (13)2 y

13

+ A4ξminus (52)

K321 K5d

minus (12)2 y

12

+ A5 K21(1 + ξ + η) + K2ξ1113960

+ K561 K3ξ

minus (56)d

minus (16)2 + K

531 K4ξ

minus (53)d

minus (12)2

+ K521 K5ξ

minus (52)d

minus (12)2 1113961

(80)



11139461

0d2Px( 1113857

2dx 11139461

0d2Qx( 1113857

2dx 11139461

0d2Rx( 1113857

2dxle 1113958M2 tge 1113957τ2

(81)




16 Complexity

ux

vx

wx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ J

minus 1

Px

Qx

Rx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

J

Pu Pv Pw

Qu Qv Qw

Ru Rv Rw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(82)


d2ux

11138681113868111386811138681113868111386811138681113868 + d2vx

11138681113868111386811138681113868111386811138681113868 + d2wx

11138681113868111386811138681113868111386811138681113868leL d2Px

11138681113868111386811138681113868111386811138681113868 + d2Qx

11138681113868111386811138681113868111386811138681113868 + d2Rx

111386811138681113868111386811138681113868111386811138681113872 1113873

0lt xlt 1 tgt 0

(83)



11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w





11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w














2d1 + 2α11M + α12M + α13M( 1113857

+ λ1113957vM2 α131113957u + ρα31 1113957w( 1113857

2d2 + α21M + 2α22M + α23M( 1113857

+ ρ1113957wM2 α121113957u + λα211113957v( 1113857

2d3 + α31M + α32M + 2α33M( 1113857

(87)



Define

V(u v w) 11139461


u


v

1113957v1113874 11138751113882


1113957w1113874 11138751113883dx

(88)


dV(u v w)

dt minus 1113946

1

0

1113957u

u2 d1 + p11( 1113857u2x +

λ1113957v

v2d2 + p22( 1113857v

2x1113896

+ρ1113957w

w2 d3 + p33( 1113857w2x +

α121113957u

u+λα211113957v

v1113888 1113889uxvx

+α131113957u

u+ρα31 1113957w

w1113874 1113875uxwx

+λα231113957v

v+ρα32 1113957w

w1113888 1113889vxwx1113897dx

minus 11139461

0

v

1113957uu(u minus 1113957u)

2+

bλu

1113957vv(v minus 1113957v)

21113896

+ ρr(w minus 1113957w)2

minusgλ + 1



(89)

Complexity 17


4λρ1113957u1113957v1113957w d1 + p11( 1113857 d2 + p22( 1113857 d3 + p33( 1113857


gt ρ1113957w α121113957uv + λα211113957vu( 11138572

d3 + p33( 1113857

+ λ1113957v α131113957uw + ρα31 1113957wu( 11138572

d2 + p22( 1113857

+ 1113957u λα231113957vw + ρα32 1113957wv( 11138572

d1 + p11( 1113857

(90)



1

0(u minus 1113957u)

2+(v minus 1113957v)

2+(w minus 1113957w)

21113960 1113961dx

dH(u v w)

dtlt 0 (u v w)neE3

(91)


10[(u minus 1113957u)2 + (v minus 1113957v)2 +





t⟶infin

(92)





t⟶infin

(93)



α11 2

α12 1

α13 10

α21 32

α22 12

α23 1

α31 10

α32 2

α33 3

a 12

b 12

g 1

s 3

e 1

r 5

d1 d2 d3≫ 1

(94)


Data Availability




18 Complexity

Acknowledgments


References



























Complexity 19



|u|41)leCM20(|ux|22 + M2


a 11139461

0u2vdxle a 1113946

1

0

u4

2εdx + a 1113946

1

0

ε2v2dx

a2CM2

02dlowast

11139461

0v2dx +

dlowast

211139461

0u2xdx +

dlowastM20

2

(60)


11139461

0u2

+ v2

+ w2


211139461

0u2x + v

2x + w

2x1113872 1113873dx

+ C2 11139461

0u2

+ v2

+ w2

1113872 1113873dx +dlowastM2

02

(61)


|u|61 leCM40(|ux|22 + M2

0) erefore

minus 11139461

0u2x + v

2x + w

2x1113872 1113873dxle 3M

20 minus

19CM2

011139461

0u2

+ v2

+ w2

1113872 1113873dx1113890 1113891

3

(62)


12ddt

11139461

0u2

+ v2

+ w2


11139461

0u2

+ v2

+ w2

1113872 1113873dx1113890 1113891

3

+ C2 11139461

0u2

+ v2

+ w2

1113872 1113873dx + 2dlowastM

20

(63)


such that

11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w





11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w



1113957u u

d2

1113957v v

d2

1113957w w

d2

1113957t d1t

(66)






u(x 0) u0(x)

v(x 0) v0(x)

w(x 0) w0(x)

0ltxlt 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(67)




1u minus gdminus 1




11139461

0udx 1113946

1

0vdx 1113946

1

0wdxleM0d

minus 12

11139461

0u2dx 1113946

1

0v2dx 1113946

1

0w

2dxleM1dminus 22


(68)


1


1

0v2tdx

minus η11139461

0w


1


+ 11139461


+ ξ11139461

0p21utk + 1+ p22( 1113857vtk + p23wtk1113858 1113859dx

+ 11139461


(69)

Complexity 11

where y 111393810(P2

x + Q2x + R2


1 2 3) such that


2t + w

2t1113872 1113873 (70)

erefore


1


1


1

0w


1

0(u + v + w) u

2t + v

2t + w

2t1113872 1113873dx

+ 11139461

01 + p11ξ( 1113857utfdx + 1113946

1

0ξ 1 + p22( 1113857vtkdx + 1113946

1

0η + p33ξ( 1113857wthdx

+ 11139461

0p12ξvtfdx + 1113946

1

0p13ξwtfdx + 1113946

1

0p21ξutkdx

+ 11139461

0p23ξwtkdx + 1113946

1

0p31ξuthdx + 1113946

1

0p32ξvthdx

(71)


11139461

0u4dxleM

131 d

minus (23)1 1113946

1

0u5dx1113888 1113889

23

11139461

0u2v2dxleM

131 d

minus (23)1 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

11139461

0u3dxleM

231 d

minus (43)1 1113946

1

0u5dx1113888 1113889

13

11139461

0uv

2dxleM231 d

minus (43)1 1113946

1

0v5dx1113888 1113889

13

11139461

0u3wdxleM

131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

12

11139461

0w

5dx1113888 1113889

16

11139461

0uvwdxleM

231 d

minus (43)2 1113946

1

0u5dx1113888 1113889

16

11139461

0v5dx1113888 1113889

16

11139461

0u2vwdxleM

131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

16

(72)



12 Complexity

minus 11139461

0u2tdxle minus

12

11139461

0P2xxdx + 1113946

1

0f2dx

minus ξ 11139461

0v2tdxle minus

ξ2

11139461

0Q

2xxdx + ξ 1113946

1

0k2dx

minus η11139461

0w

2tdxle minus

η2

11139461

0R2xxdx + η1113946

1

0h2dx

11139461

0f2dxle 2M1d

minus 21 d

minus 22 + a

2ξ2M131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

+ ξ2M131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0w

5dx1113888 1113889

13

+ 2aξdminus 11 d

minus (43)2 M

231 1113946

1

0v5dx1113888 1113889

13

+ 2ξdminus 11 d

minus (23)2 M

231 1113946

1

0u5dx1113888 1113889

16

11139461

0v5dx1113888 1113889

16


131 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

16

ξ 11139461

0k2dxle ξd

minus 21 d

minus 22 M1 b

2+ g

21113872 1113873 + ξ3dminus (23)

2 M131 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

+ ξ2gdminus 11 d

minus (43)2 M

231 1113946

1

0v5dx1113888 1113889

13

η11139461

0h2dxle ηs

2d

minus 21 d

minus 22 M1 + ηd

2ξ2dminus (23)2 M

131 1113946

1

0u5dx1113888 1113889

13

ηr2ξ2dminus (23)

2 M131


2 M131 1113946

1

0u5dx1113888 1113889

23


131 1113946

1

0u5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

12

(73)

us

minus 11139461


1


1

0w

2tdxle minus

12

11139461

0P2xxdx minus

ξ2

11139461

0Q

2xxdx minus

η2

11139461

0R2xxdx + C5(1 + ξ + η)d

minus 21 d

minus 22 M1

+ C6dminus (43)2 M

231 ξ(1 + η) 1113946

1

0u5

+ v5

1113872 1113873dx1113888 1113889

13

+ C7dminus (23)2 M

131 ξ2(1 + η) 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx1113888 1113889

23

(74)

Complexity 13

11139461

0utfdxle d

minus 11 1113946

1

0uutdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ d

minus 11 1113946

1

0utvdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ aξ 1113946

1

0uutvdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ ξ 1113946

1

0uutwdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le dminus 11

12ϵ

11139461

0udx +ϵ2

11139461

0uu

2tdx1113888 1113889 + d

minus 11

12ϵ

11139461

0vdx +ϵ2

11139461

0vu

2tdx1113888 1113889

+ aξ12ϵ

11139461

0uv

2dx +ϵ2

11139461

0uu

2tdx1113888 1113889 + ξ

12ϵ

11139461

0uw

2dx +ϵ2

11139461

0uu

2tdx1113888 1113889


2ϵ

M0 +a

2ϵM

231 d

minus (43)2 ξ 1113946

1

0v5dx1113888 1113889

13

+ξ2ϵ

M231 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+ϵdminus 1

12

11139461

0vu

2tdx +ϵ2

dminus 11 + aξ + ξ1113872 1113873 1113946

1

0uu

2tdx

2α11ξ 11139461

0uutfdxle

α11ϵ

M231 d

minus 11 d

232 1113946

1

0u5dx1113888 1113889

13

+ 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦ +

3α11(a + 1)

5ϵξ2 1113946

1

0u5dx

+2α11a5ϵ

ξ2 11139461

0v5dx +

2α115ϵ

ξ2 11139461

0w

5dx + α11ξϵ 2dminus 11 + aξ + ξ1113872 1113873 1113946

1

0uu

2tdx

2α12ξ 11139461

0vutfdxle

α12ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0v5dx1113888 1113889

13

+α12(a + 1)

5ϵξ2 1113946

1

0u5dx +

2α125ϵ

ξ2 11139461

0w

5dx +α12(3a + 4)

10ϵξ2 1113946

1

0v5dx

α13ξ 11139461

0wutfdxle

α13ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α13(a +(12))

5ϵξ2 1113946

1

0u5dx +

2aα135ϵ

ξ2 11139461

0v5dx

+2α13(a + 1)

5ϵξ2 1113946

1

0w

5dx +α13dminus 1

12ϵξ 1113946

1

0vu

2tdx +ϵ2


2+ 11113872 1113873 1113946

1

0uu

2tdx

ξ 11139461

0vtkdxle

b + g

2ϵM0d

minus 11 d

minus 12 ξ +

ξ2

2ϵM

231 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+bdminus 1

12ϵξ 1113946

1

0uv

2tdx +ϵ2ξ gd

minus 11 + ξ1113872 1113873 1113946

1

0vv

2tdx

α21ξ 11139461

0uvtkdxle

α21(b + g)

2ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+2α215ϵ

ξ2 11139461

0u5dx

+α2110ϵ

ξ2 11139461

0v5dx +

α21bdminus 11

2ϵξ 1113946

1

0uv

2tdx +

α21ϵξ2

gdminus 11 + ξ1113872 1113873 1113946

1

0vv

2tdx

2α22ξ 11139461

0vvtk dx le

α22(b + g)

ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+α225ϵ

ξ2 11139461

0u5dx

+4α225ϵ

ξ2 11139461

0v5dx + α22(b + g)d

minus 11 ϵξ 1113946

1

0vv

2tdx + α22ϵξ

211139461

0uv

2tdx

α23ξ 11139461

0wvtkdxle

α23(b + g)

2ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

minus (43)

+α23ϵξ2

bdminus 11 + ξ1113872 1113873 1113946

1

0uv

2tdx

+α235ϵ

ξ2 11139461

0u5dx + 21113946

1

0v5dx + 21113946

1

0w

5dx1113888 1113889 +α23gdminus 1

12ϵξ 1113946

1

0vv

2tdx

η11139461

0wthdxle

s

ϵd

minus 11 d

minus 12 M0 +

r

2ϵM

231 d

minus (43)2 ηξ 1113946

1

0w

5dx1113888 1113889

13

+e

2ϵM

231 d

minus (43)2 ηξ 1113946

1

0w

5dx1113888 1113889

13ηϵ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

α31ξ 11139461

0uwthdxle

α31s2ϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0u5dx1113888 1113889

13

+α31(r + 2e)

5ϵξ2 1113946

1

0u5dx1113888 1113889

13

+α31(3r + e)

10ϵξ2 1113946

1

0w

5dx1113888 1113889

13

+α31ξϵ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

14 Complexity

α32ξ 11139461

0vwthdxle

α32s2ϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0v5dx1113888 1113889

13

+2α32e5ϵ

ξ2 11139461

0u5dx +

α32(r + 2e)

5ϵξ2 1113946

1

0v5dx

+α32(e +(3r2))

5ϵξ2 1113946

1

0w

5dx +α32ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

2α33ξ 11139461

0wwthdxle

α33sϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0w

5dx1113888 1113889

13

+2α33e5ϵ

ξ2

+α33(5r + 3e)

5ϵξ2 1113946

1

0w


1

0ww

2tdx

α12ξ 11139461

0uvtfdxle

α12ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+α125ϵ

2a +32

1113874 1113875ξ2 11139461

0u5dx

+aα1210ϵ

ξ2 11139461

0v5dx +

α125ϵ

ξ2 11139461

0w

5dx +α12ϵξ2

dminus 11 + ξ1113872 1113873 1113946

1

0uv

2tdx +

α12ϵξ2

dminus 11 + aξ1113872 1113873 1113946

1

0vv

2tdx

α12ξ 11139461

0uwtfdxle

α12ξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+ 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+3α12(a + 1)

10ϵξ2 1113946

1

0u5dx +

aα125ϵ

ξ2 11139461

0v5dx +

α125ϵ

ξ2 11139461

0w


aξ2

+ξ2

1113888 1113889 11139461

0uw

2tdx

α21ξ 11139461

0vutkdxle

α21ξ2ϵ

M231 d

minus 11 d

minus (43)2 b 1113946

1

0u5dx1113888 1113889

13

+ g 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+α215ϵ

ξ2 11139461

0u5dx +

3α2110ϵ

ξ2 11139461

0v5dx +

α21ϵξ2

bdminus 11 + gd

minus 11 + ξ1113872 1113873 1113946

1

0vu

2tdx

α23ξ 11139461

0vwtkdxle

α23ξ2ϵ

M231 d

minus 11 d

minus (43)2 b 1113946

1

0u5dx1113888 1113889

13

+ g 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+α235ϵ

ξ2 11139461

0u5dx +

3α2310ϵ

ξ2 11139461

0v5dx +

α23ϵξ2

bdminus 11 + gd

minus 11 + ξ1113872 1113873 1113946

1

0vw

2tdx

α31ξ 11139461

0wuthdxle

α31sξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α31e5ϵ

ξ2 11139461

0u5dx

+α31ξ

2

2ϵr +

35

e1113874 1113875 11139461

0w

5dx +α31ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0wu

2tdx

α32ξ 11139461

0wvthdxle

α32sξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α32e5ϵ

ξ2 11139461

0u5dx

+α32ξ

2

2ϵr +

35

e1113874 1113875 11139461

0w

5dx +α32ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0wv

2tdx

(75)

Complexity 15

erefore

11139461

01 + p11ξ( 1113857utfdx + ξ 1113946

1

01 + p22( 1113857vtkdx + 1113946

1

0η + p33ξ( 1113857wthdx

+ ξ 11139461

0p12vtfdx + ξ 1113946

1

0p13wtfdx + ξ 1113946

1

0p21utkdx

+ ξ 11139461

0p23wtkdx + ξ 1113946

1

0p31uthdx + ξ 1113946

1

0p32vthdx

le λϵξ 11139461

0(u + v + w) u

2t + v

2t + w

2t1113872 1113873dx +

C8

ϵM0d

minus 11 d

minus 22 (1 + ξ + η)

+C9

ϵM

231 d

minus (43)2 ξ 1 + d

minus 11 + ξ + η1113872 1113873 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx1113890 1113891

13

+C10

ϵξ2 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx

(76)



12

11139461

0P2xxdx minus

ξ2

11139461

0Q

2xxdx minus

η2

11139461

0R2xxdx

+ B1K2dminus 11 d

minus 12 + B2K3d

minus (43)1 z

13


23+ B4K5z

(77)

where z 111393810(u5 + v5 + w5)dx K2 (1 + ξ + η)


2 ) K3 M231 ξ(1 + dminus 1

1 + ξ + η)



zleB5ξminus (52)

11139461

0P52

+ Q52

+ R52

1113872 1113873dx

leB6ξminus (52)

K321 d

minus (32)1 y

12+ B6ξ

minus (52)K

521 d

minus (52)1

z13 leB7ξ

minus (56)K

121 d

minus (12)1 y

16+ B7ξ

minus (56)K

561 d

minus (56)1

z23 leB8ξ

minus (53)K1d

minus 11 y

13+ B8ξ

minus (53)K

531 d

minus (53)1

(78)


431 d

minus (43)1 (|Pxx|22 + K2

1dminus 21 ) So

minusξ2

1113946

1

0

P2xxdx minus

12

1113946

1

0

Q2xxdx minus

η2

1113946

1

0

R2xxdx


431 y

53+(1 + ξ + η)K

21d

minus 21

(79)


2 we have

12


432 y

53

+ A2ξminus (56)

K121 K3d

minus (16)2 y

16

+ A3ξminus (53)

K1K4dminus (13)2 y

13

+ A4ξminus (52)

K321 K5d

minus (12)2 y

12

+ A5 K21(1 + ξ + η) + K2ξ1113960

+ K561 K3ξ

minus (56)d

minus (16)2 + K

531 K4ξ

minus (53)d

minus (12)2

+ K521 K5ξ

minus (52)d

minus (12)2 1113961

(80)



11139461

0d2Px( 1113857

2dx 11139461

0d2Qx( 1113857

2dx 11139461

0d2Rx( 1113857

2dxle 1113958M2 tge 1113957τ2

(81)




16 Complexity

ux

vx

wx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ J

minus 1

Px

Qx

Rx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

J

Pu Pv Pw

Qu Qv Qw

Ru Rv Rw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(82)


d2ux

11138681113868111386811138681113868111386811138681113868 + d2vx

11138681113868111386811138681113868111386811138681113868 + d2wx

11138681113868111386811138681113868111386811138681113868leL d2Px

11138681113868111386811138681113868111386811138681113868 + d2Qx

11138681113868111386811138681113868111386811138681113868 + d2Rx

111386811138681113868111386811138681113868111386811138681113872 1113873

0lt xlt 1 tgt 0

(83)



11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w





11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w














2d1 + 2α11M + α12M + α13M( 1113857

+ λ1113957vM2 α131113957u + ρα31 1113957w( 1113857

2d2 + α21M + 2α22M + α23M( 1113857

+ ρ1113957wM2 α121113957u + λα211113957v( 1113857

2d3 + α31M + α32M + 2α33M( 1113857

(87)



Define

V(u v w) 11139461


u


v

1113957v1113874 11138751113882


1113957w1113874 11138751113883dx

(88)


dV(u v w)

dt minus 1113946

1

0

1113957u

u2 d1 + p11( 1113857u2x +

λ1113957v

v2d2 + p22( 1113857v

2x1113896

+ρ1113957w

w2 d3 + p33( 1113857w2x +

α121113957u

u+λα211113957v

v1113888 1113889uxvx

+α131113957u

u+ρα31 1113957w

w1113874 1113875uxwx

+λα231113957v

v+ρα32 1113957w

w1113888 1113889vxwx1113897dx

minus 11139461

0

v

1113957uu(u minus 1113957u)

2+

bλu

1113957vv(v minus 1113957v)

21113896

+ ρr(w minus 1113957w)2

minusgλ + 1



(89)

Complexity 17


4λρ1113957u1113957v1113957w d1 + p11( 1113857 d2 + p22( 1113857 d3 + p33( 1113857


gt ρ1113957w α121113957uv + λα211113957vu( 11138572

d3 + p33( 1113857

+ λ1113957v α131113957uw + ρα31 1113957wu( 11138572

d2 + p22( 1113857

+ 1113957u λα231113957vw + ρα32 1113957wv( 11138572

d1 + p11( 1113857

(90)



1

0(u minus 1113957u)

2+(v minus 1113957v)

2+(w minus 1113957w)

21113960 1113961dx

dH(u v w)

dtlt 0 (u v w)neE3

(91)


10[(u minus 1113957u)2 + (v minus 1113957v)2 +





t⟶infin

(92)





t⟶infin

(93)



α11 2

α12 1

α13 10

α21 32

α22 12

α23 1

α31 10

α32 2

α33 3

a 12

b 12

g 1

s 3

e 1

r 5

d1 d2 d3≫ 1

(94)


Data Availability




18 Complexity

Acknowledgments


References



























Complexity 19

where y 111393810(P2

x + Q2x + R2


1 2 3) such that


2t + w

2t1113872 1113873 (70)

erefore


1


1


1

0w


1

0(u + v + w) u

2t + v

2t + w

2t1113872 1113873dx

+ 11139461

01 + p11ξ( 1113857utfdx + 1113946

1

0ξ 1 + p22( 1113857vtkdx + 1113946

1

0η + p33ξ( 1113857wthdx

+ 11139461

0p12ξvtfdx + 1113946

1

0p13ξwtfdx + 1113946

1

0p21ξutkdx

+ 11139461

0p23ξwtkdx + 1113946

1

0p31ξuthdx + 1113946

1

0p32ξvthdx

(71)


11139461

0u4dxleM

131 d

minus (23)1 1113946

1

0u5dx1113888 1113889

23

11139461

0u2v2dxleM

131 d

minus (23)1 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

11139461

0u3dxleM

231 d

minus (43)1 1113946

1

0u5dx1113888 1113889

13

11139461

0uv

2dxleM231 d

minus (43)1 1113946

1

0v5dx1113888 1113889

13

11139461

0u3wdxleM

131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

12

11139461

0w

5dx1113888 1113889

16

11139461

0uvwdxleM

231 d

minus (43)2 1113946

1

0u5dx1113888 1113889

16

11139461

0v5dx1113888 1113889

16

11139461

0u2vwdxleM

131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

16

(72)



12 Complexity

minus 11139461

0u2tdxle minus

12

11139461

0P2xxdx + 1113946

1

0f2dx

minus ξ 11139461

0v2tdxle minus

ξ2

11139461

0Q

2xxdx + ξ 1113946

1

0k2dx

minus η11139461

0w

2tdxle minus

η2

11139461

0R2xxdx + η1113946

1

0h2dx

11139461

0f2dxle 2M1d

minus 21 d

minus 22 + a

2ξ2M131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

+ ξ2M131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0w

5dx1113888 1113889

13

+ 2aξdminus 11 d

minus (43)2 M

231 1113946

1

0v5dx1113888 1113889

13

+ 2ξdminus 11 d

minus (23)2 M

231 1113946

1

0u5dx1113888 1113889

16

11139461

0v5dx1113888 1113889

16


131 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

16

ξ 11139461

0k2dxle ξd

minus 21 d

minus 22 M1 b

2+ g

21113872 1113873 + ξ3dminus (23)

2 M131 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

+ ξ2gdminus 11 d

minus (43)2 M

231 1113946

1

0v5dx1113888 1113889

13

η11139461

0h2dxle ηs

2d

minus 21 d

minus 22 M1 + ηd

2ξ2dminus (23)2 M

131 1113946

1

0u5dx1113888 1113889

13

ηr2ξ2dminus (23)

2 M131


2 M131 1113946

1

0u5dx1113888 1113889

23


131 1113946

1

0u5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

12

(73)

us

minus 11139461


1


1

0w

2tdxle minus

12

11139461

0P2xxdx minus

ξ2

11139461

0Q

2xxdx minus

η2

11139461

0R2xxdx + C5(1 + ξ + η)d

minus 21 d

minus 22 M1

+ C6dminus (43)2 M

231 ξ(1 + η) 1113946

1

0u5

+ v5

1113872 1113873dx1113888 1113889

13

+ C7dminus (23)2 M

131 ξ2(1 + η) 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx1113888 1113889

23

(74)

Complexity 13

11139461

0utfdxle d

minus 11 1113946

1

0uutdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ d

minus 11 1113946

1

0utvdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ aξ 1113946

1

0uutvdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ ξ 1113946

1

0uutwdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le dminus 11

12ϵ

11139461

0udx +ϵ2

11139461

0uu

2tdx1113888 1113889 + d

minus 11

12ϵ

11139461

0vdx +ϵ2

11139461

0vu

2tdx1113888 1113889

+ aξ12ϵ

11139461

0uv

2dx +ϵ2

11139461

0uu

2tdx1113888 1113889 + ξ

12ϵ

11139461

0uw

2dx +ϵ2

11139461

0uu

2tdx1113888 1113889


2ϵ

M0 +a

2ϵM

231 d

minus (43)2 ξ 1113946

1

0v5dx1113888 1113889

13

+ξ2ϵ

M231 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+ϵdminus 1

12

11139461

0vu

2tdx +ϵ2

dminus 11 + aξ + ξ1113872 1113873 1113946

1

0uu

2tdx

2α11ξ 11139461

0uutfdxle

α11ϵ

M231 d

minus 11 d

232 1113946

1

0u5dx1113888 1113889

13

+ 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦ +

3α11(a + 1)

5ϵξ2 1113946

1

0u5dx

+2α11a5ϵ

ξ2 11139461

0v5dx +

2α115ϵ

ξ2 11139461

0w

5dx + α11ξϵ 2dminus 11 + aξ + ξ1113872 1113873 1113946

1

0uu

2tdx

2α12ξ 11139461

0vutfdxle

α12ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0v5dx1113888 1113889

13

+α12(a + 1)

5ϵξ2 1113946

1

0u5dx +

2α125ϵ

ξ2 11139461

0w

5dx +α12(3a + 4)

10ϵξ2 1113946

1

0v5dx

α13ξ 11139461

0wutfdxle

α13ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α13(a +(12))

5ϵξ2 1113946

1

0u5dx +

2aα135ϵ

ξ2 11139461

0v5dx

+2α13(a + 1)

5ϵξ2 1113946

1

0w

5dx +α13dminus 1

12ϵξ 1113946

1

0vu

2tdx +ϵ2


2+ 11113872 1113873 1113946

1

0uu

2tdx

ξ 11139461

0vtkdxle

b + g

2ϵM0d

minus 11 d

minus 12 ξ +

ξ2

2ϵM

231 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+bdminus 1

12ϵξ 1113946

1

0uv

2tdx +ϵ2ξ gd

minus 11 + ξ1113872 1113873 1113946

1

0vv

2tdx

α21ξ 11139461

0uvtkdxle

α21(b + g)

2ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+2α215ϵ

ξ2 11139461

0u5dx

+α2110ϵ

ξ2 11139461

0v5dx +

α21bdminus 11

2ϵξ 1113946

1

0uv

2tdx +

α21ϵξ2

gdminus 11 + ξ1113872 1113873 1113946

1

0vv

2tdx

2α22ξ 11139461

0vvtk dx le

α22(b + g)

ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+α225ϵ

ξ2 11139461

0u5dx

+4α225ϵ

ξ2 11139461

0v5dx + α22(b + g)d

minus 11 ϵξ 1113946

1

0vv

2tdx + α22ϵξ

211139461

0uv

2tdx

α23ξ 11139461

0wvtkdxle

α23(b + g)

2ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

minus (43)

+α23ϵξ2

bdminus 11 + ξ1113872 1113873 1113946

1

0uv

2tdx

+α235ϵ

ξ2 11139461

0u5dx + 21113946

1

0v5dx + 21113946

1

0w

5dx1113888 1113889 +α23gdminus 1

12ϵξ 1113946

1

0vv

2tdx

η11139461

0wthdxle

s

ϵd

minus 11 d

minus 12 M0 +

r

2ϵM

231 d

minus (43)2 ηξ 1113946

1

0w

5dx1113888 1113889

13

+e

2ϵM

231 d

minus (43)2 ηξ 1113946

1

0w

5dx1113888 1113889

13ηϵ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

α31ξ 11139461

0uwthdxle

α31s2ϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0u5dx1113888 1113889

13

+α31(r + 2e)

5ϵξ2 1113946

1

0u5dx1113888 1113889

13

+α31(3r + e)

10ϵξ2 1113946

1

0w

5dx1113888 1113889

13

+α31ξϵ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

14 Complexity

α32ξ 11139461

0vwthdxle

α32s2ϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0v5dx1113888 1113889

13

+2α32e5ϵ

ξ2 11139461

0u5dx +

α32(r + 2e)

5ϵξ2 1113946

1

0v5dx

+α32(e +(3r2))

5ϵξ2 1113946

1

0w

5dx +α32ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

2α33ξ 11139461

0wwthdxle

α33sϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0w

5dx1113888 1113889

13

+2α33e5ϵ

ξ2

+α33(5r + 3e)

5ϵξ2 1113946

1

0w


1

0ww

2tdx

α12ξ 11139461

0uvtfdxle

α12ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+α125ϵ

2a +32

1113874 1113875ξ2 11139461

0u5dx

+aα1210ϵ

ξ2 11139461

0v5dx +

α125ϵ

ξ2 11139461

0w

5dx +α12ϵξ2

dminus 11 + ξ1113872 1113873 1113946

1

0uv

2tdx +

α12ϵξ2

dminus 11 + aξ1113872 1113873 1113946

1

0vv

2tdx

α12ξ 11139461

0uwtfdxle

α12ξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+ 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+3α12(a + 1)

10ϵξ2 1113946

1

0u5dx +

aα125ϵ

ξ2 11139461

0v5dx +

α125ϵ

ξ2 11139461

0w


aξ2

+ξ2

1113888 1113889 11139461

0uw

2tdx

α21ξ 11139461

0vutkdxle

α21ξ2ϵ

M231 d

minus 11 d

minus (43)2 b 1113946

1

0u5dx1113888 1113889

13

+ g 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+α215ϵ

ξ2 11139461

0u5dx +

3α2110ϵ

ξ2 11139461

0v5dx +

α21ϵξ2

bdminus 11 + gd

minus 11 + ξ1113872 1113873 1113946

1

0vu

2tdx

α23ξ 11139461

0vwtkdxle

α23ξ2ϵ

M231 d

minus 11 d

minus (43)2 b 1113946

1

0u5dx1113888 1113889

13

+ g 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+α235ϵ

ξ2 11139461

0u5dx +

3α2310ϵ

ξ2 11139461

0v5dx +

α23ϵξ2

bdminus 11 + gd

minus 11 + ξ1113872 1113873 1113946

1

0vw

2tdx

α31ξ 11139461

0wuthdxle

α31sξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α31e5ϵ

ξ2 11139461

0u5dx

+α31ξ

2

2ϵr +

35

e1113874 1113875 11139461

0w

5dx +α31ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0wu

2tdx

α32ξ 11139461

0wvthdxle

α32sξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α32e5ϵ

ξ2 11139461

0u5dx

+α32ξ

2

2ϵr +

35

e1113874 1113875 11139461

0w

5dx +α32ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0wv

2tdx

(75)

Complexity 15

erefore

11139461

01 + p11ξ( 1113857utfdx + ξ 1113946

1

01 + p22( 1113857vtkdx + 1113946

1

0η + p33ξ( 1113857wthdx

+ ξ 11139461

0p12vtfdx + ξ 1113946

1

0p13wtfdx + ξ 1113946

1

0p21utkdx

+ ξ 11139461

0p23wtkdx + ξ 1113946

1

0p31uthdx + ξ 1113946

1

0p32vthdx

le λϵξ 11139461

0(u + v + w) u

2t + v

2t + w

2t1113872 1113873dx +

C8

ϵM0d

minus 11 d

minus 22 (1 + ξ + η)

+C9

ϵM

231 d

minus (43)2 ξ 1 + d

minus 11 + ξ + η1113872 1113873 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx1113890 1113891

13

+C10

ϵξ2 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx

(76)



12

11139461

0P2xxdx minus

ξ2

11139461

0Q

2xxdx minus

η2

11139461

0R2xxdx

+ B1K2dminus 11 d

minus 12 + B2K3d

minus (43)1 z

13


23+ B4K5z

(77)

where z 111393810(u5 + v5 + w5)dx K2 (1 + ξ + η)


2 ) K3 M231 ξ(1 + dminus 1

1 + ξ + η)



zleB5ξminus (52)

11139461

0P52

+ Q52

+ R52

1113872 1113873dx

leB6ξminus (52)

K321 d

minus (32)1 y

12+ B6ξ

minus (52)K

521 d

minus (52)1

z13 leB7ξ

minus (56)K

121 d

minus (12)1 y

16+ B7ξ

minus (56)K

561 d

minus (56)1

z23 leB8ξ

minus (53)K1d

minus 11 y

13+ B8ξ

minus (53)K

531 d

minus (53)1

(78)


431 d

minus (43)1 (|Pxx|22 + K2

1dminus 21 ) So

minusξ2

1113946

1

0

P2xxdx minus

12

1113946

1

0

Q2xxdx minus

η2

1113946

1

0

R2xxdx


431 y

53+(1 + ξ + η)K

21d

minus 21

(79)


2 we have

12


432 y

53

+ A2ξminus (56)

K121 K3d

minus (16)2 y

16

+ A3ξminus (53)

K1K4dminus (13)2 y

13

+ A4ξminus (52)

K321 K5d

minus (12)2 y

12

+ A5 K21(1 + ξ + η) + K2ξ1113960

+ K561 K3ξ

minus (56)d

minus (16)2 + K

531 K4ξ

minus (53)d

minus (12)2

+ K521 K5ξ

minus (52)d

minus (12)2 1113961

(80)



11139461

0d2Px( 1113857

2dx 11139461

0d2Qx( 1113857

2dx 11139461

0d2Rx( 1113857

2dxle 1113958M2 tge 1113957τ2

(81)




16 Complexity

ux

vx

wx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ J

minus 1

Px

Qx

Rx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

J

Pu Pv Pw

Qu Qv Qw

Ru Rv Rw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(82)


d2ux

11138681113868111386811138681113868111386811138681113868 + d2vx

11138681113868111386811138681113868111386811138681113868 + d2wx

11138681113868111386811138681113868111386811138681113868leL d2Px

11138681113868111386811138681113868111386811138681113868 + d2Qx

11138681113868111386811138681113868111386811138681113868 + d2Rx

111386811138681113868111386811138681113868111386811138681113872 1113873

0lt xlt 1 tgt 0

(83)



11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w





11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w














2d1 + 2α11M + α12M + α13M( 1113857

+ λ1113957vM2 α131113957u + ρα31 1113957w( 1113857

2d2 + α21M + 2α22M + α23M( 1113857

+ ρ1113957wM2 α121113957u + λα211113957v( 1113857

2d3 + α31M + α32M + 2α33M( 1113857

(87)



Define

V(u v w) 11139461


u


v

1113957v1113874 11138751113882


1113957w1113874 11138751113883dx

(88)


dV(u v w)

dt minus 1113946

1

0

1113957u

u2 d1 + p11( 1113857u2x +

λ1113957v

v2d2 + p22( 1113857v

2x1113896

+ρ1113957w

w2 d3 + p33( 1113857w2x +

α121113957u

u+λα211113957v

v1113888 1113889uxvx

+α131113957u

u+ρα31 1113957w

w1113874 1113875uxwx

+λα231113957v

v+ρα32 1113957w

w1113888 1113889vxwx1113897dx

minus 11139461

0

v

1113957uu(u minus 1113957u)

2+

bλu

1113957vv(v minus 1113957v)

21113896

+ ρr(w minus 1113957w)2

minusgλ + 1



(89)

Complexity 17


4λρ1113957u1113957v1113957w d1 + p11( 1113857 d2 + p22( 1113857 d3 + p33( 1113857


gt ρ1113957w α121113957uv + λα211113957vu( 11138572

d3 + p33( 1113857

+ λ1113957v α131113957uw + ρα31 1113957wu( 11138572

d2 + p22( 1113857

+ 1113957u λα231113957vw + ρα32 1113957wv( 11138572

d1 + p11( 1113857

(90)



1

0(u minus 1113957u)

2+(v minus 1113957v)

2+(w minus 1113957w)

21113960 1113961dx

dH(u v w)

dtlt 0 (u v w)neE3

(91)


10[(u minus 1113957u)2 + (v minus 1113957v)2 +





t⟶infin

(92)





t⟶infin

(93)



α11 2

α12 1

α13 10

α21 32

α22 12

α23 1

α31 10

α32 2

α33 3

a 12

b 12

g 1

s 3

e 1

r 5

d1 d2 d3≫ 1

(94)


Data Availability




18 Complexity

Acknowledgments


References



























Complexity 19

minus 11139461

0u2tdxle minus

12

11139461

0P2xxdx + 1113946

1

0f2dx

minus ξ 11139461

0v2tdxle minus

ξ2

11139461

0Q

2xxdx + ξ 1113946

1

0k2dx

minus η11139461

0w

2tdxle minus

η2

11139461

0R2xxdx + η1113946

1

0h2dx

11139461

0f2dxle 2M1d

minus 21 d

minus 22 + a

2ξ2M131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

+ ξ2M131 d

minus (23)2 1113946

1

0u5dx1113888 1113889

13

11139461

0w

5dx1113888 1113889

13

+ 2aξdminus 11 d

minus (43)2 M

231 1113946

1

0v5dx1113888 1113889

13

+ 2ξdminus 11 d

minus (23)2 M

231 1113946

1

0u5dx1113888 1113889

16

11139461

0v5dx1113888 1113889

16


131 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

16

ξ 11139461

0k2dxle ξd

minus 21 d

minus 22 M1 b

2+ g

21113872 1113873 + ξ3dminus (23)

2 M131 1113946

1

0u5dx1113888 1113889

13

11139461

0v5dx1113888 1113889

13

+ ξ2gdminus 11 d

minus (43)2 M

231 1113946

1

0v5dx1113888 1113889

13

η11139461

0h2dxle ηs

2d

minus 21 d

minus 22 M1 + ηd

2ξ2dminus (23)2 M

131 1113946

1

0u5dx1113888 1113889

13

ηr2ξ2dminus (23)

2 M131


2 M131 1113946

1

0u5dx1113888 1113889

23


131 1113946

1

0u5dx1113888 1113889

16

11139461

0w

5dx1113888 1113889

12

(73)

us

minus 11139461


1


1

0w

2tdxle minus

12

11139461

0P2xxdx minus

ξ2

11139461

0Q

2xxdx minus

η2

11139461

0R2xxdx + C5(1 + ξ + η)d

minus 21 d

minus 22 M1

+ C6dminus (43)2 M

231 ξ(1 + η) 1113946

1

0u5

+ v5

1113872 1113873dx1113888 1113889

13

+ C7dminus (23)2 M

131 ξ2(1 + η) 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx1113888 1113889

23

(74)

Complexity 13

11139461

0utfdxle d

minus 11 1113946

1

0uutdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ d

minus 11 1113946

1

0utvdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ aξ 1113946

1

0uutvdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ ξ 1113946

1

0uutwdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le dminus 11

12ϵ

11139461

0udx +ϵ2

11139461

0uu

2tdx1113888 1113889 + d

minus 11

12ϵ

11139461

0vdx +ϵ2

11139461

0vu

2tdx1113888 1113889

+ aξ12ϵ

11139461

0uv

2dx +ϵ2

11139461

0uu

2tdx1113888 1113889 + ξ

12ϵ

11139461

0uw

2dx +ϵ2

11139461

0uu

2tdx1113888 1113889


2ϵ

M0 +a

2ϵM

231 d

minus (43)2 ξ 1113946

1

0v5dx1113888 1113889

13

+ξ2ϵ

M231 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+ϵdminus 1

12

11139461

0vu

2tdx +ϵ2

dminus 11 + aξ + ξ1113872 1113873 1113946

1

0uu

2tdx

2α11ξ 11139461

0uutfdxle

α11ϵ

M231 d

minus 11 d

232 1113946

1

0u5dx1113888 1113889

13

+ 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦ +

3α11(a + 1)

5ϵξ2 1113946

1

0u5dx

+2α11a5ϵ

ξ2 11139461

0v5dx +

2α115ϵ

ξ2 11139461

0w

5dx + α11ξϵ 2dminus 11 + aξ + ξ1113872 1113873 1113946

1

0uu

2tdx

2α12ξ 11139461

0vutfdxle

α12ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0v5dx1113888 1113889

13

+α12(a + 1)

5ϵξ2 1113946

1

0u5dx +

2α125ϵ

ξ2 11139461

0w

5dx +α12(3a + 4)

10ϵξ2 1113946

1

0v5dx

α13ξ 11139461

0wutfdxle

α13ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α13(a +(12))

5ϵξ2 1113946

1

0u5dx +

2aα135ϵ

ξ2 11139461

0v5dx

+2α13(a + 1)

5ϵξ2 1113946

1

0w

5dx +α13dminus 1

12ϵξ 1113946

1

0vu

2tdx +ϵ2


2+ 11113872 1113873 1113946

1

0uu

2tdx

ξ 11139461

0vtkdxle

b + g

2ϵM0d

minus 11 d

minus 12 ξ +

ξ2

2ϵM

231 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+bdminus 1

12ϵξ 1113946

1

0uv

2tdx +ϵ2ξ gd

minus 11 + ξ1113872 1113873 1113946

1

0vv

2tdx

α21ξ 11139461

0uvtkdxle

α21(b + g)

2ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+2α215ϵ

ξ2 11139461

0u5dx

+α2110ϵ

ξ2 11139461

0v5dx +

α21bdminus 11

2ϵξ 1113946

1

0uv

2tdx +

α21ϵξ2

gdminus 11 + ξ1113872 1113873 1113946

1

0vv

2tdx

2α22ξ 11139461

0vvtk dx le

α22(b + g)

ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+α225ϵ

ξ2 11139461

0u5dx

+4α225ϵ

ξ2 11139461

0v5dx + α22(b + g)d

minus 11 ϵξ 1113946

1

0vv

2tdx + α22ϵξ

211139461

0uv

2tdx

α23ξ 11139461

0wvtkdxle

α23(b + g)

2ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

minus (43)

+α23ϵξ2

bdminus 11 + ξ1113872 1113873 1113946

1

0uv

2tdx

+α235ϵ

ξ2 11139461

0u5dx + 21113946

1

0v5dx + 21113946

1

0w

5dx1113888 1113889 +α23gdminus 1

12ϵξ 1113946

1

0vv

2tdx

η11139461

0wthdxle

s

ϵd

minus 11 d

minus 12 M0 +

r

2ϵM

231 d

minus (43)2 ηξ 1113946

1

0w

5dx1113888 1113889

13

+e

2ϵM

231 d

minus (43)2 ηξ 1113946

1

0w

5dx1113888 1113889

13ηϵ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

α31ξ 11139461

0uwthdxle

α31s2ϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0u5dx1113888 1113889

13

+α31(r + 2e)

5ϵξ2 1113946

1

0u5dx1113888 1113889

13

+α31(3r + e)

10ϵξ2 1113946

1

0w

5dx1113888 1113889

13

+α31ξϵ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

14 Complexity

α32ξ 11139461

0vwthdxle

α32s2ϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0v5dx1113888 1113889

13

+2α32e5ϵ

ξ2 11139461

0u5dx +

α32(r + 2e)

5ϵξ2 1113946

1

0v5dx

+α32(e +(3r2))

5ϵξ2 1113946

1

0w

5dx +α32ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

2α33ξ 11139461

0wwthdxle

α33sϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0w

5dx1113888 1113889

13

+2α33e5ϵ

ξ2

+α33(5r + 3e)

5ϵξ2 1113946

1

0w


1

0ww

2tdx

α12ξ 11139461

0uvtfdxle

α12ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+α125ϵ

2a +32

1113874 1113875ξ2 11139461

0u5dx

+aα1210ϵ

ξ2 11139461

0v5dx +

α125ϵ

ξ2 11139461

0w

5dx +α12ϵξ2

dminus 11 + ξ1113872 1113873 1113946

1

0uv

2tdx +

α12ϵξ2

dminus 11 + aξ1113872 1113873 1113946

1

0vv

2tdx

α12ξ 11139461

0uwtfdxle

α12ξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+ 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+3α12(a + 1)

10ϵξ2 1113946

1

0u5dx +

aα125ϵ

ξ2 11139461

0v5dx +

α125ϵ

ξ2 11139461

0w


aξ2

+ξ2

1113888 1113889 11139461

0uw

2tdx

α21ξ 11139461

0vutkdxle

α21ξ2ϵ

M231 d

minus 11 d

minus (43)2 b 1113946

1

0u5dx1113888 1113889

13

+ g 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+α215ϵ

ξ2 11139461

0u5dx +

3α2110ϵ

ξ2 11139461

0v5dx +

α21ϵξ2

bdminus 11 + gd

minus 11 + ξ1113872 1113873 1113946

1

0vu

2tdx

α23ξ 11139461

0vwtkdxle

α23ξ2ϵ

M231 d

minus 11 d

minus (43)2 b 1113946

1

0u5dx1113888 1113889

13

+ g 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+α235ϵ

ξ2 11139461

0u5dx +

3α2310ϵ

ξ2 11139461

0v5dx +

α23ϵξ2

bdminus 11 + gd

minus 11 + ξ1113872 1113873 1113946

1

0vw

2tdx

α31ξ 11139461

0wuthdxle

α31sξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α31e5ϵ

ξ2 11139461

0u5dx

+α31ξ

2

2ϵr +

35

e1113874 1113875 11139461

0w

5dx +α31ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0wu

2tdx

α32ξ 11139461

0wvthdxle

α32sξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α32e5ϵ

ξ2 11139461

0u5dx

+α32ξ

2

2ϵr +

35

e1113874 1113875 11139461

0w

5dx +α32ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0wv

2tdx

(75)

Complexity 15

erefore

11139461

01 + p11ξ( 1113857utfdx + ξ 1113946

1

01 + p22( 1113857vtkdx + 1113946

1

0η + p33ξ( 1113857wthdx

+ ξ 11139461

0p12vtfdx + ξ 1113946

1

0p13wtfdx + ξ 1113946

1

0p21utkdx

+ ξ 11139461

0p23wtkdx + ξ 1113946

1

0p31uthdx + ξ 1113946

1

0p32vthdx

le λϵξ 11139461

0(u + v + w) u

2t + v

2t + w

2t1113872 1113873dx +

C8

ϵM0d

minus 11 d

minus 22 (1 + ξ + η)

+C9

ϵM

231 d

minus (43)2 ξ 1 + d

minus 11 + ξ + η1113872 1113873 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx1113890 1113891

13

+C10

ϵξ2 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx

(76)



12

11139461

0P2xxdx minus

ξ2

11139461

0Q

2xxdx minus

η2

11139461

0R2xxdx

+ B1K2dminus 11 d

minus 12 + B2K3d

minus (43)1 z

13


23+ B4K5z

(77)

where z 111393810(u5 + v5 + w5)dx K2 (1 + ξ + η)


2 ) K3 M231 ξ(1 + dminus 1

1 + ξ + η)



zleB5ξminus (52)

11139461

0P52

+ Q52

+ R52

1113872 1113873dx

leB6ξminus (52)

K321 d

minus (32)1 y

12+ B6ξ

minus (52)K

521 d

minus (52)1

z13 leB7ξ

minus (56)K

121 d

minus (12)1 y

16+ B7ξ

minus (56)K

561 d

minus (56)1

z23 leB8ξ

minus (53)K1d

minus 11 y

13+ B8ξ

minus (53)K

531 d

minus (53)1

(78)


431 d

minus (43)1 (|Pxx|22 + K2

1dminus 21 ) So

minusξ2

1113946

1

0

P2xxdx minus

12

1113946

1

0

Q2xxdx minus

η2

1113946

1

0

R2xxdx


431 y

53+(1 + ξ + η)K

21d

minus 21

(79)


2 we have

12


432 y

53

+ A2ξminus (56)

K121 K3d

minus (16)2 y

16

+ A3ξminus (53)

K1K4dminus (13)2 y

13

+ A4ξminus (52)

K321 K5d

minus (12)2 y

12

+ A5 K21(1 + ξ + η) + K2ξ1113960

+ K561 K3ξ

minus (56)d

minus (16)2 + K

531 K4ξ

minus (53)d

minus (12)2

+ K521 K5ξ

minus (52)d

minus (12)2 1113961

(80)



11139461

0d2Px( 1113857

2dx 11139461

0d2Qx( 1113857

2dx 11139461

0d2Rx( 1113857

2dxle 1113958M2 tge 1113957τ2

(81)




16 Complexity

ux

vx

wx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ J

minus 1

Px

Qx

Rx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

J

Pu Pv Pw

Qu Qv Qw

Ru Rv Rw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(82)


d2ux

11138681113868111386811138681113868111386811138681113868 + d2vx

11138681113868111386811138681113868111386811138681113868 + d2wx

11138681113868111386811138681113868111386811138681113868leL d2Px

11138681113868111386811138681113868111386811138681113868 + d2Qx

11138681113868111386811138681113868111386811138681113868 + d2Rx

111386811138681113868111386811138681113868111386811138681113872 1113873

0lt xlt 1 tgt 0

(83)



11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w





11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w














2d1 + 2α11M + α12M + α13M( 1113857

+ λ1113957vM2 α131113957u + ρα31 1113957w( 1113857

2d2 + α21M + 2α22M + α23M( 1113857

+ ρ1113957wM2 α121113957u + λα211113957v( 1113857

2d3 + α31M + α32M + 2α33M( 1113857

(87)



Define

V(u v w) 11139461


u


v

1113957v1113874 11138751113882


1113957w1113874 11138751113883dx

(88)


dV(u v w)

dt minus 1113946

1

0

1113957u

u2 d1 + p11( 1113857u2x +

λ1113957v

v2d2 + p22( 1113857v

2x1113896

+ρ1113957w

w2 d3 + p33( 1113857w2x +

α121113957u

u+λα211113957v

v1113888 1113889uxvx

+α131113957u

u+ρα31 1113957w

w1113874 1113875uxwx

+λα231113957v

v+ρα32 1113957w

w1113888 1113889vxwx1113897dx

minus 11139461

0

v

1113957uu(u minus 1113957u)

2+

bλu

1113957vv(v minus 1113957v)

21113896

+ ρr(w minus 1113957w)2

minusgλ + 1



(89)

Complexity 17


4λρ1113957u1113957v1113957w d1 + p11( 1113857 d2 + p22( 1113857 d3 + p33( 1113857


gt ρ1113957w α121113957uv + λα211113957vu( 11138572

d3 + p33( 1113857

+ λ1113957v α131113957uw + ρα31 1113957wu( 11138572

d2 + p22( 1113857

+ 1113957u λα231113957vw + ρα32 1113957wv( 11138572

d1 + p11( 1113857

(90)



1

0(u minus 1113957u)

2+(v minus 1113957v)

2+(w minus 1113957w)

21113960 1113961dx

dH(u v w)

dtlt 0 (u v w)neE3

(91)


10[(u minus 1113957u)2 + (v minus 1113957v)2 +





t⟶infin

(92)





t⟶infin

(93)



α11 2

α12 1

α13 10

α21 32

α22 12

α23 1

α31 10

α32 2

α33 3

a 12

b 12

g 1

s 3

e 1

r 5

d1 d2 d3≫ 1

(94)


Data Availability




18 Complexity

Acknowledgments


References



























Complexity 19

11139461

0utfdxle d

minus 11 1113946

1

0uutdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ d

minus 11 1113946

1

0utvdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ aξ 1113946

1

0uutvdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ ξ 1113946

1

0uutwdx

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le dminus 11

12ϵ

11139461

0udx +ϵ2

11139461

0uu

2tdx1113888 1113889 + d

minus 11

12ϵ

11139461

0vdx +ϵ2

11139461

0vu

2tdx1113888 1113889

+ aξ12ϵ

11139461

0uv

2dx +ϵ2

11139461

0uu

2tdx1113888 1113889 + ξ

12ϵ

11139461

0uw

2dx +ϵ2

11139461

0uu

2tdx1113888 1113889


2ϵ

M0 +a

2ϵM

231 d

minus (43)2 ξ 1113946

1

0v5dx1113888 1113889

13

+ξ2ϵ

M231 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+ϵdminus 1

12

11139461

0vu

2tdx +ϵ2

dminus 11 + aξ + ξ1113872 1113873 1113946

1

0uu

2tdx

2α11ξ 11139461

0uutfdxle

α11ϵ

M231 d

minus 11 d

232 1113946

1

0u5dx1113888 1113889

13

+ 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦ +

3α11(a + 1)

5ϵξ2 1113946

1

0u5dx

+2α11a5ϵ

ξ2 11139461

0v5dx +

2α115ϵ

ξ2 11139461

0w

5dx + α11ξϵ 2dminus 11 + aξ + ξ1113872 1113873 1113946

1

0uu

2tdx

2α12ξ 11139461

0vutfdxle

α12ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0v5dx1113888 1113889

13

+α12(a + 1)

5ϵξ2 1113946

1

0u5dx +

2α125ϵ

ξ2 11139461

0w

5dx +α12(3a + 4)

10ϵξ2 1113946

1

0v5dx

α13ξ 11139461

0wutfdxle

α13ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α13(a +(12))

5ϵξ2 1113946

1

0u5dx +

2aα135ϵ

ξ2 11139461

0v5dx

+2α13(a + 1)

5ϵξ2 1113946

1

0w

5dx +α13dminus 1

12ϵξ 1113946

1

0vu

2tdx +ϵ2


2+ 11113872 1113873 1113946

1

0uu

2tdx

ξ 11139461

0vtkdxle

b + g

2ϵM0d

minus 11 d

minus 12 ξ +

ξ2

2ϵM

231 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+bdminus 1

12ϵξ 1113946

1

0uv

2tdx +ϵ2ξ gd

minus 11 + ξ1113872 1113873 1113946

1

0vv

2tdx

α21ξ 11139461

0uvtkdxle

α21(b + g)

2ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+2α215ϵ

ξ2 11139461

0u5dx

+α2110ϵ

ξ2 11139461

0v5dx +

α21bdminus 11

2ϵξ 1113946

1

0uv

2tdx +

α21ϵξ2

gdminus 11 + ξ1113872 1113873 1113946

1

0vv

2tdx

2α22ξ 11139461

0vvtk dx le

α22(b + g)

ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+α225ϵ

ξ2 11139461

0u5dx

+4α225ϵ

ξ2 11139461

0v5dx + α22(b + g)d

minus 11 ϵξ 1113946

1

0vv

2tdx + α22ϵξ

211139461

0uv

2tdx

α23ξ 11139461

0wvtkdxle

α23(b + g)

2ϵM

231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

minus (43)

+α23ϵξ2

bdminus 11 + ξ1113872 1113873 1113946

1

0uv

2tdx

+α235ϵ

ξ2 11139461

0u5dx + 21113946

1

0v5dx + 21113946

1

0w

5dx1113888 1113889 +α23gdminus 1

12ϵξ 1113946

1

0vv

2tdx

η11139461

0wthdxle

s

ϵd

minus 11 d

minus 12 M0 +

r

2ϵM

231 d

minus (43)2 ηξ 1113946

1

0w

5dx1113888 1113889

13

+e

2ϵM

231 d

minus (43)2 ηξ 1113946

1

0w

5dx1113888 1113889

13ηϵ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

α31ξ 11139461

0uwthdxle

α31s2ϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0u5dx1113888 1113889

13

+α31(r + 2e)

5ϵξ2 1113946

1

0u5dx1113888 1113889

13

+α31(3r + e)

10ϵξ2 1113946

1

0w

5dx1113888 1113889

13

+α31ξϵ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

14 Complexity

α32ξ 11139461

0vwthdxle

α32s2ϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0v5dx1113888 1113889

13

+2α32e5ϵ

ξ2 11139461

0u5dx +

α32(r + 2e)

5ϵξ2 1113946

1

0v5dx

+α32(e +(3r2))

5ϵξ2 1113946

1

0w

5dx +α32ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

2α33ξ 11139461

0wwthdxle

α33sϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0w

5dx1113888 1113889

13

+2α33e5ϵ

ξ2

+α33(5r + 3e)

5ϵξ2 1113946

1

0w


1

0ww

2tdx

α12ξ 11139461

0uvtfdxle

α12ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+α125ϵ

2a +32

1113874 1113875ξ2 11139461

0u5dx

+aα1210ϵ

ξ2 11139461

0v5dx +

α125ϵ

ξ2 11139461

0w

5dx +α12ϵξ2

dminus 11 + ξ1113872 1113873 1113946

1

0uv

2tdx +

α12ϵξ2

dminus 11 + aξ1113872 1113873 1113946

1

0vv

2tdx

α12ξ 11139461

0uwtfdxle

α12ξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+ 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+3α12(a + 1)

10ϵξ2 1113946

1

0u5dx +

aα125ϵ

ξ2 11139461

0v5dx +

α125ϵ

ξ2 11139461

0w


aξ2

+ξ2

1113888 1113889 11139461

0uw

2tdx

α21ξ 11139461

0vutkdxle

α21ξ2ϵ

M231 d

minus 11 d

minus (43)2 b 1113946

1

0u5dx1113888 1113889

13

+ g 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+α215ϵ

ξ2 11139461

0u5dx +

3α2110ϵ

ξ2 11139461

0v5dx +

α21ϵξ2

bdminus 11 + gd

minus 11 + ξ1113872 1113873 1113946

1

0vu

2tdx

α23ξ 11139461

0vwtkdxle

α23ξ2ϵ

M231 d

minus 11 d

minus (43)2 b 1113946

1

0u5dx1113888 1113889

13

+ g 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+α235ϵ

ξ2 11139461

0u5dx +

3α2310ϵ

ξ2 11139461

0v5dx +

α23ϵξ2

bdminus 11 + gd

minus 11 + ξ1113872 1113873 1113946

1

0vw

2tdx

α31ξ 11139461

0wuthdxle

α31sξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α31e5ϵ

ξ2 11139461

0u5dx

+α31ξ

2

2ϵr +

35

e1113874 1113875 11139461

0w

5dx +α31ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0wu

2tdx

α32ξ 11139461

0wvthdxle

α32sξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α32e5ϵ

ξ2 11139461

0u5dx

+α32ξ

2

2ϵr +

35

e1113874 1113875 11139461

0w

5dx +α32ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0wv

2tdx

(75)

Complexity 15

erefore

11139461

01 + p11ξ( 1113857utfdx + ξ 1113946

1

01 + p22( 1113857vtkdx + 1113946

1

0η + p33ξ( 1113857wthdx

+ ξ 11139461

0p12vtfdx + ξ 1113946

1

0p13wtfdx + ξ 1113946

1

0p21utkdx

+ ξ 11139461

0p23wtkdx + ξ 1113946

1

0p31uthdx + ξ 1113946

1

0p32vthdx

le λϵξ 11139461

0(u + v + w) u

2t + v

2t + w

2t1113872 1113873dx +

C8

ϵM0d

minus 11 d

minus 22 (1 + ξ + η)

+C9

ϵM

231 d

minus (43)2 ξ 1 + d

minus 11 + ξ + η1113872 1113873 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx1113890 1113891

13

+C10

ϵξ2 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx

(76)



12

11139461

0P2xxdx minus

ξ2

11139461

0Q

2xxdx minus

η2

11139461

0R2xxdx

+ B1K2dminus 11 d

minus 12 + B2K3d

minus (43)1 z

13


23+ B4K5z

(77)

where z 111393810(u5 + v5 + w5)dx K2 (1 + ξ + η)


2 ) K3 M231 ξ(1 + dminus 1

1 + ξ + η)



zleB5ξminus (52)

11139461

0P52

+ Q52

+ R52

1113872 1113873dx

leB6ξminus (52)

K321 d

minus (32)1 y

12+ B6ξ

minus (52)K

521 d

minus (52)1

z13 leB7ξ

minus (56)K

121 d

minus (12)1 y

16+ B7ξ

minus (56)K

561 d

minus (56)1

z23 leB8ξ

minus (53)K1d

minus 11 y

13+ B8ξ

minus (53)K

531 d

minus (53)1

(78)


431 d

minus (43)1 (|Pxx|22 + K2

1dminus 21 ) So

minusξ2

1113946

1

0

P2xxdx minus

12

1113946

1

0

Q2xxdx minus

η2

1113946

1

0

R2xxdx


431 y

53+(1 + ξ + η)K

21d

minus 21

(79)


2 we have

12


432 y

53

+ A2ξminus (56)

K121 K3d

minus (16)2 y

16

+ A3ξminus (53)

K1K4dminus (13)2 y

13

+ A4ξminus (52)

K321 K5d

minus (12)2 y

12

+ A5 K21(1 + ξ + η) + K2ξ1113960

+ K561 K3ξ

minus (56)d

minus (16)2 + K

531 K4ξ

minus (53)d

minus (12)2

+ K521 K5ξ

minus (52)d

minus (12)2 1113961

(80)



11139461

0d2Px( 1113857

2dx 11139461

0d2Qx( 1113857

2dx 11139461

0d2Rx( 1113857

2dxle 1113958M2 tge 1113957τ2

(81)




16 Complexity

ux

vx

wx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ J

minus 1

Px

Qx

Rx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

J

Pu Pv Pw

Qu Qv Qw

Ru Rv Rw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(82)


d2ux

11138681113868111386811138681113868111386811138681113868 + d2vx

11138681113868111386811138681113868111386811138681113868 + d2wx

11138681113868111386811138681113868111386811138681113868leL d2Px

11138681113868111386811138681113868111386811138681113868 + d2Qx

11138681113868111386811138681113868111386811138681113868 + d2Rx

111386811138681113868111386811138681113868111386811138681113872 1113873

0lt xlt 1 tgt 0

(83)



11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w





11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w














2d1 + 2α11M + α12M + α13M( 1113857

+ λ1113957vM2 α131113957u + ρα31 1113957w( 1113857

2d2 + α21M + 2α22M + α23M( 1113857

+ ρ1113957wM2 α121113957u + λα211113957v( 1113857

2d3 + α31M + α32M + 2α33M( 1113857

(87)



Define

V(u v w) 11139461


u


v

1113957v1113874 11138751113882


1113957w1113874 11138751113883dx

(88)


dV(u v w)

dt minus 1113946

1

0

1113957u

u2 d1 + p11( 1113857u2x +

λ1113957v

v2d2 + p22( 1113857v

2x1113896

+ρ1113957w

w2 d3 + p33( 1113857w2x +

α121113957u

u+λα211113957v

v1113888 1113889uxvx

+α131113957u

u+ρα31 1113957w

w1113874 1113875uxwx

+λα231113957v

v+ρα32 1113957w

w1113888 1113889vxwx1113897dx

minus 11139461

0

v

1113957uu(u minus 1113957u)

2+

bλu

1113957vv(v minus 1113957v)

21113896

+ ρr(w minus 1113957w)2

minusgλ + 1



(89)

Complexity 17


4λρ1113957u1113957v1113957w d1 + p11( 1113857 d2 + p22( 1113857 d3 + p33( 1113857


gt ρ1113957w α121113957uv + λα211113957vu( 11138572

d3 + p33( 1113857

+ λ1113957v α131113957uw + ρα31 1113957wu( 11138572

d2 + p22( 1113857

+ 1113957u λα231113957vw + ρα32 1113957wv( 11138572

d1 + p11( 1113857

(90)



1

0(u minus 1113957u)

2+(v minus 1113957v)

2+(w minus 1113957w)

21113960 1113961dx

dH(u v w)

dtlt 0 (u v w)neE3

(91)


10[(u minus 1113957u)2 + (v minus 1113957v)2 +





t⟶infin

(92)





t⟶infin

(93)



α11 2

α12 1

α13 10

α21 32

α22 12

α23 1

α31 10

α32 2

α33 3

a 12

b 12

g 1

s 3

e 1

r 5

d1 d2 d3≫ 1

(94)


Data Availability




18 Complexity

Acknowledgments


References



























Complexity 19

α32ξ 11139461

0vwthdxle

α32s2ϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0v5dx1113888 1113889

13

+2α32e5ϵ

ξ2 11139461

0u5dx +

α32(r + 2e)

5ϵξ2 1113946

1

0v5dx

+α32(e +(3r2))

5ϵξ2 1113946

1

0w

5dx +α32ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0ww

2tdx

2α33ξ 11139461

0wwthdxle

α33sϵ

M231 d

minus 11 d

minus (43)2 ξ 1113946

1

0w

5dx1113888 1113889

13

+2α33e5ϵ

ξ2

+α33(5r + 3e)

5ϵξ2 1113946

1

0w


1

0ww

2tdx

α12ξ 11139461

0uvtfdxle

α12ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+α125ϵ

2a +32

1113874 1113875ξ2 11139461

0u5dx

+aα1210ϵ

ξ2 11139461

0v5dx +

α125ϵ

ξ2 11139461

0w

5dx +α12ϵξ2

dminus 11 + ξ1113872 1113873 1113946

1

0uv

2tdx +

α12ϵξ2

dminus 11 + aξ1113872 1113873 1113946

1

0vv

2tdx

α12ξ 11139461

0uwtfdxle

α12ξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0u5dx1113888 1113889

13

+ 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+3α12(a + 1)

10ϵξ2 1113946

1

0u5dx +

aα125ϵ

ξ2 11139461

0v5dx +

α125ϵ

ξ2 11139461

0w


aξ2

+ξ2

1113888 1113889 11139461

0uw

2tdx

α21ξ 11139461

0vutkdxle

α21ξ2ϵ

M231 d

minus 11 d

minus (43)2 b 1113946

1

0u5dx1113888 1113889

13

+ g 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+α215ϵ

ξ2 11139461

0u5dx +

3α2110ϵ

ξ2 11139461

0v5dx +

α21ϵξ2

bdminus 11 + gd

minus 11 + ξ1113872 1113873 1113946

1

0vu

2tdx

α23ξ 11139461

0vwtkdxle

α23ξ2ϵ

M231 d

minus 11 d

minus (43)2 b 1113946

1

0u5dx1113888 1113889

13

+ g 11139461

0v5dx1113888 1113889

13⎡⎣ ⎤⎦

+α235ϵ

ξ2 11139461

0u5dx +

3α2310ϵ

ξ2 11139461

0v5dx +

α23ϵξ2

bdminus 11 + gd

minus 11 + ξ1113872 1113873 1113946

1

0vw

2tdx

α31ξ 11139461

0wuthdxle

α31sξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α31e5ϵ

ξ2 11139461

0u5dx

+α31ξ

2

2ϵr +

35

e1113874 1113875 11139461

0w

5dx +α31ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0wu

2tdx

α32ξ 11139461

0wvthdxle

α32sξ2ϵ

M231 d

minus 11 d

minus (43)2 1113946

1

0w

5dx1113888 1113889

13

+α32e5ϵ

ξ2 11139461

0u5dx

+α32ξ

2

2ϵr +

35

e1113874 1113875 11139461

0w

5dx +α32ϵξ2

sdminus 11 + rξ + eξ1113872 1113873 1113946

1

0wv

2tdx

(75)

Complexity 15

erefore

11139461

01 + p11ξ( 1113857utfdx + ξ 1113946

1

01 + p22( 1113857vtkdx + 1113946

1

0η + p33ξ( 1113857wthdx

+ ξ 11139461

0p12vtfdx + ξ 1113946

1

0p13wtfdx + ξ 1113946

1

0p21utkdx

+ ξ 11139461

0p23wtkdx + ξ 1113946

1

0p31uthdx + ξ 1113946

1

0p32vthdx

le λϵξ 11139461

0(u + v + w) u

2t + v

2t + w

2t1113872 1113873dx +

C8

ϵM0d

minus 11 d

minus 22 (1 + ξ + η)

+C9

ϵM

231 d

minus (43)2 ξ 1 + d

minus 11 + ξ + η1113872 1113873 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx1113890 1113891

13

+C10

ϵξ2 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx

(76)



12

11139461

0P2xxdx minus

ξ2

11139461

0Q

2xxdx minus

η2

11139461

0R2xxdx

+ B1K2dminus 11 d

minus 12 + B2K3d

minus (43)1 z

13


23+ B4K5z

(77)

where z 111393810(u5 + v5 + w5)dx K2 (1 + ξ + η)


2 ) K3 M231 ξ(1 + dminus 1

1 + ξ + η)



zleB5ξminus (52)

11139461

0P52

+ Q52

+ R52

1113872 1113873dx

leB6ξminus (52)

K321 d

minus (32)1 y

12+ B6ξ

minus (52)K

521 d

minus (52)1

z13 leB7ξ

minus (56)K

121 d

minus (12)1 y

16+ B7ξ

minus (56)K

561 d

minus (56)1

z23 leB8ξ

minus (53)K1d

minus 11 y

13+ B8ξ

minus (53)K

531 d

minus (53)1

(78)


431 d

minus (43)1 (|Pxx|22 + K2

1dminus 21 ) So

minusξ2

1113946

1

0

P2xxdx minus

12

1113946

1

0

Q2xxdx minus

η2

1113946

1

0

R2xxdx


431 y

53+(1 + ξ + η)K

21d

minus 21

(79)


2 we have

12


432 y

53

+ A2ξminus (56)

K121 K3d

minus (16)2 y

16

+ A3ξminus (53)

K1K4dminus (13)2 y

13

+ A4ξminus (52)

K321 K5d

minus (12)2 y

12

+ A5 K21(1 + ξ + η) + K2ξ1113960

+ K561 K3ξ

minus (56)d

minus (16)2 + K

531 K4ξ

minus (53)d

minus (12)2

+ K521 K5ξ

minus (52)d

minus (12)2 1113961

(80)



11139461

0d2Px( 1113857

2dx 11139461

0d2Qx( 1113857

2dx 11139461

0d2Rx( 1113857

2dxle 1113958M2 tge 1113957τ2

(81)




16 Complexity

ux

vx

wx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ J

minus 1

Px

Qx

Rx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

J

Pu Pv Pw

Qu Qv Qw

Ru Rv Rw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(82)


d2ux

11138681113868111386811138681113868111386811138681113868 + d2vx

11138681113868111386811138681113868111386811138681113868 + d2wx

11138681113868111386811138681113868111386811138681113868leL d2Px

11138681113868111386811138681113868111386811138681113868 + d2Qx

11138681113868111386811138681113868111386811138681113868 + d2Rx

111386811138681113868111386811138681113868111386811138681113872 1113873

0lt xlt 1 tgt 0

(83)



11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w





11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w














2d1 + 2α11M + α12M + α13M( 1113857

+ λ1113957vM2 α131113957u + ρα31 1113957w( 1113857

2d2 + α21M + 2α22M + α23M( 1113857

+ ρ1113957wM2 α121113957u + λα211113957v( 1113857

2d3 + α31M + α32M + 2α33M( 1113857

(87)



Define

V(u v w) 11139461


u


v

1113957v1113874 11138751113882


1113957w1113874 11138751113883dx

(88)


dV(u v w)

dt minus 1113946

1

0

1113957u

u2 d1 + p11( 1113857u2x +

λ1113957v

v2d2 + p22( 1113857v

2x1113896

+ρ1113957w

w2 d3 + p33( 1113857w2x +

α121113957u

u+λα211113957v

v1113888 1113889uxvx

+α131113957u

u+ρα31 1113957w

w1113874 1113875uxwx

+λα231113957v

v+ρα32 1113957w

w1113888 1113889vxwx1113897dx

minus 11139461

0

v

1113957uu(u minus 1113957u)

2+

bλu

1113957vv(v minus 1113957v)

21113896

+ ρr(w minus 1113957w)2

minusgλ + 1



(89)

Complexity 17


4λρ1113957u1113957v1113957w d1 + p11( 1113857 d2 + p22( 1113857 d3 + p33( 1113857


gt ρ1113957w α121113957uv + λα211113957vu( 11138572

d3 + p33( 1113857

+ λ1113957v α131113957uw + ρα31 1113957wu( 11138572

d2 + p22( 1113857

+ 1113957u λα231113957vw + ρα32 1113957wv( 11138572

d1 + p11( 1113857

(90)



1

0(u minus 1113957u)

2+(v minus 1113957v)

2+(w minus 1113957w)

21113960 1113961dx

dH(u v w)

dtlt 0 (u v w)neE3

(91)


10[(u minus 1113957u)2 + (v minus 1113957v)2 +





t⟶infin

(92)





t⟶infin

(93)



α11 2

α12 1

α13 10

α21 32

α22 12

α23 1

α31 10

α32 2

α33 3

a 12

b 12

g 1

s 3

e 1

r 5

d1 d2 d3≫ 1

(94)


Data Availability




18 Complexity

Acknowledgments


References



























Complexity 19

erefore

11139461

01 + p11ξ( 1113857utfdx + ξ 1113946

1

01 + p22( 1113857vtkdx + 1113946

1

0η + p33ξ( 1113857wthdx

+ ξ 11139461

0p12vtfdx + ξ 1113946

1

0p13wtfdx + ξ 1113946

1

0p21utkdx

+ ξ 11139461

0p23wtkdx + ξ 1113946

1

0p31uthdx + ξ 1113946

1

0p32vthdx

le λϵξ 11139461

0(u + v + w) u

2t + v

2t + w

2t1113872 1113873dx +

C8

ϵM0d

minus 11 d

minus 22 (1 + ξ + η)

+C9

ϵM

231 d

minus (43)2 ξ 1 + d

minus 11 + ξ + η1113872 1113873 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx1113890 1113891

13

+C10

ϵξ2 1113946

1

0u5

+ v5

+ w5

1113872 1113873dx

(76)



12

11139461

0P2xxdx minus

ξ2

11139461

0Q

2xxdx minus

η2

11139461

0R2xxdx

+ B1K2dminus 11 d

minus 12 + B2K3d

minus (43)1 z

13


23+ B4K5z

(77)

where z 111393810(u5 + v5 + w5)dx K2 (1 + ξ + η)


2 ) K3 M231 ξ(1 + dminus 1

1 + ξ + η)



zleB5ξminus (52)

11139461

0P52

+ Q52

+ R52

1113872 1113873dx

leB6ξminus (52)

K321 d

minus (32)1 y

12+ B6ξ

minus (52)K

521 d

minus (52)1

z13 leB7ξ

minus (56)K

121 d

minus (12)1 y

16+ B7ξ

minus (56)K

561 d

minus (56)1

z23 leB8ξ

minus (53)K1d

minus 11 y

13+ B8ξ

minus (53)K

531 d

minus (53)1

(78)


431 d

minus (43)1 (|Pxx|22 + K2

1dminus 21 ) So

minusξ2

1113946

1

0

P2xxdx minus

12

1113946

1

0

Q2xxdx minus

η2

1113946

1

0

R2xxdx


431 y

53+(1 + ξ + η)K

21d

minus 21

(79)


2 we have

12


432 y

53

+ A2ξminus (56)

K121 K3d

minus (16)2 y

16

+ A3ξminus (53)

K1K4dminus (13)2 y

13

+ A4ξminus (52)

K321 K5d

minus (12)2 y

12

+ A5 K21(1 + ξ + η) + K2ξ1113960

+ K561 K3ξ

minus (56)d

minus (16)2 + K

531 K4ξ

minus (53)d

minus (12)2

+ K521 K5ξ

minus (52)d

minus (12)2 1113961

(80)



11139461

0d2Px( 1113857

2dx 11139461

0d2Qx( 1113857

2dx 11139461

0d2Rx( 1113857

2dxle 1113958M2 tge 1113957τ2

(81)




16 Complexity

ux

vx

wx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ J

minus 1

Px

Qx

Rx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

J

Pu Pv Pw

Qu Qv Qw

Ru Rv Rw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(82)


d2ux

11138681113868111386811138681113868111386811138681113868 + d2vx

11138681113868111386811138681113868111386811138681113868 + d2wx

11138681113868111386811138681113868111386811138681113868leL d2Px

11138681113868111386811138681113868111386811138681113868 + d2Qx

11138681113868111386811138681113868111386811138681113868 + d2Rx

111386811138681113868111386811138681113868111386811138681113872 1113873

0lt xlt 1 tgt 0

(83)



11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w





11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w














2d1 + 2α11M + α12M + α13M( 1113857

+ λ1113957vM2 α131113957u + ρα31 1113957w( 1113857

2d2 + α21M + 2α22M + α23M( 1113857

+ ρ1113957wM2 α121113957u + λα211113957v( 1113857

2d3 + α31M + α32M + 2α33M( 1113857

(87)



Define

V(u v w) 11139461


u


v

1113957v1113874 11138751113882


1113957w1113874 11138751113883dx

(88)


dV(u v w)

dt minus 1113946

1

0

1113957u

u2 d1 + p11( 1113857u2x +

λ1113957v

v2d2 + p22( 1113857v

2x1113896

+ρ1113957w

w2 d3 + p33( 1113857w2x +

α121113957u

u+λα211113957v

v1113888 1113889uxvx

+α131113957u

u+ρα31 1113957w

w1113874 1113875uxwx

+λα231113957v

v+ρα32 1113957w

w1113888 1113889vxwx1113897dx

minus 11139461

0

v

1113957uu(u minus 1113957u)

2+

bλu

1113957vv(v minus 1113957v)

21113896

+ ρr(w minus 1113957w)2

minusgλ + 1



(89)

Complexity 17


4λρ1113957u1113957v1113957w d1 + p11( 1113857 d2 + p22( 1113857 d3 + p33( 1113857


gt ρ1113957w α121113957uv + λα211113957vu( 11138572

d3 + p33( 1113857

+ λ1113957v α131113957uw + ρα31 1113957wu( 11138572

d2 + p22( 1113857

+ 1113957u λα231113957vw + ρα32 1113957wv( 11138572

d1 + p11( 1113857

(90)



1

0(u minus 1113957u)

2+(v minus 1113957v)

2+(w minus 1113957w)

21113960 1113961dx

dH(u v w)

dtlt 0 (u v w)neE3

(91)


10[(u minus 1113957u)2 + (v minus 1113957v)2 +





t⟶infin

(92)





t⟶infin

(93)



α11 2

α12 1

α13 10

α21 32

α22 12

α23 1

α31 10

α32 2

α33 3

a 12

b 12

g 1

s 3

e 1

r 5

d1 d2 d3≫ 1

(94)


Data Availability




18 Complexity

Acknowledgments


References



























Complexity 19

ux

vx

wx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ J

minus 1

Px

Qx

Rx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

J

Pu Pv Pw

Qu Qv Qw

Ru Rv Rw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(82)


d2ux

11138681113868111386811138681113868111386811138681113868 + d2vx

11138681113868111386811138681113868111386811138681113868 + d2wx

11138681113868111386811138681113868111386811138681113868leL d2Px

11138681113868111386811138681113868111386811138681113868 + d2Qx

11138681113868111386811138681113868111386811138681113868 + d2Rx

111386811138681113868111386811138681113868111386811138681113872 1113873

0lt xlt 1 tgt 0

(83)



11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w





11139461

0u2xdx 1113946

1

0v2xdx 1113946

1

0w














2d1 + 2α11M + α12M + α13M( 1113857

+ λ1113957vM2 α131113957u + ρα31 1113957w( 1113857

2d2 + α21M + 2α22M + α23M( 1113857

+ ρ1113957wM2 α121113957u + λα211113957v( 1113857

2d3 + α31M + α32M + 2α33M( 1113857

(87)



Define

V(u v w) 11139461


u


v

1113957v1113874 11138751113882


1113957w1113874 11138751113883dx

(88)


dV(u v w)

dt minus 1113946

1

0

1113957u

u2 d1 + p11( 1113857u2x +

λ1113957v

v2d2 + p22( 1113857v

2x1113896

+ρ1113957w

w2 d3 + p33( 1113857w2x +

α121113957u

u+λα211113957v

v1113888 1113889uxvx

+α131113957u

u+ρα31 1113957w

w1113874 1113875uxwx

+λα231113957v

v+ρα32 1113957w

w1113888 1113889vxwx1113897dx

minus 11139461

0

v

1113957uu(u minus 1113957u)

2+

bλu

1113957vv(v minus 1113957v)

21113896

+ ρr(w minus 1113957w)2

minusgλ + 1



(89)

Complexity 17


4λρ1113957u1113957v1113957w d1 + p11( 1113857 d2 + p22( 1113857 d3 + p33( 1113857


gt ρ1113957w α121113957uv + λα211113957vu( 11138572

d3 + p33( 1113857

+ λ1113957v α131113957uw + ρα31 1113957wu( 11138572

d2 + p22( 1113857

+ 1113957u λα231113957vw + ρα32 1113957wv( 11138572

d1 + p11( 1113857

(90)



1

0(u minus 1113957u)

2+(v minus 1113957v)

2+(w minus 1113957w)

21113960 1113961dx

dH(u v w)

dtlt 0 (u v w)neE3

(91)


10[(u minus 1113957u)2 + (v minus 1113957v)2 +





t⟶infin

(92)





t⟶infin

(93)



α11 2

α12 1

α13 10

α21 32

α22 12

α23 1

α31 10

α32 2

α33 3

a 12

b 12

g 1

s 3

e 1

r 5

d1 d2 d3≫ 1

(94)


Data Availability




18 Complexity

Acknowledgments


References



























Complexity 19


4λρ1113957u1113957v1113957w d1 + p11( 1113857 d2 + p22( 1113857 d3 + p33( 1113857


gt ρ1113957w α121113957uv + λα211113957vu( 11138572

d3 + p33( 1113857

+ λ1113957v α131113957uw + ρα31 1113957wu( 11138572

d2 + p22( 1113857

+ 1113957u λα231113957vw + ρα32 1113957wv( 11138572

d1 + p11( 1113857

(90)



1

0(u minus 1113957u)

2+(v minus 1113957v)

2+(w minus 1113957w)

21113960 1113961dx

dH(u v w)

dtlt 0 (u v w)neE3

(91)


10[(u minus 1113957u)2 + (v minus 1113957v)2 +





t⟶infin

(92)





t⟶infin

(93)



α11 2

α12 1

α13 10

α21 32

α22 12

α23 1

α31 10

α32 2

α33 3

a 12

b 12

g 1

s 3

e 1

r 5

d1 d2 d3≫ 1

(94)


Data Availability




18 Complexity

Acknowledgments


References



























Complexity 19

Acknowledgments


References



























Complexity 19

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