research article existence of exponential -stability nonconstant...

Research ArticleExistence of Exponential 119901-Stability Nonconstant Equilibrium ofMarkovian Jumping Nonlinear Diffusion Equations via EkelandVariational Principle

Ruofeng Rao12 and Shouming Zhong3

1Department of Mathematics Chengdu Normal University Chengdu Sichuan 611130 China2Institution of Mathematics Yibin University Yibin Sichuan 644007 China3School of Mathematical Sciences University of Electronic Science and Technology of China Chengdu Sichuan 611731 China

Correspondence should be addressed to Ruofeng Rao ruofengrao163com

Received 18 February 2015 Accepted 14 June 2015

Academic Editor Klaus Kirsten

Copyright copy 2015 R Rao and S ZhongThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The authors obtained a delay-dependent exponential 119901-stability criterion for a class of Markovian jumping nonlinear diffusionequations by employing the Lyapunov stability theory and some variational methods As far as we know it is the first time to applyEkeland variational principle to obtain the existence of exponential stability equilibrium of 119901-Laplacian dynamic system so thatsome methods used in this paper are different from those methods of many previous related literatures In addition the obtainedexistence criterion is only involved in the activation functions so that the criterion is simpler and easier than other existence criteriato be verified in practical application Moreover a numerical example shows the effectiveness of the proposed methods owing tothe large allowable variation range of time-delay

1 Introduction

Nonlinear diffusion equations have been investigated exten-sively by many authors owing to their physics and biologicalengineering backgrounds population dynamics and so on(see [1ndash9] and references therein) In addition Markovianjumping systems have attracted rapidly growing interestdue to the fact that Markovian jumping parameters areuseful in modeling abrupt phenomena such as randomfailures operating in different points of a nonlinear plantand changing in the interconnections of subsystems (see[8 10ndash12] and references therein) On the other hand fuzzylogic theory has shown to be an appealing and efficientapproach to deal with the analysis and synthesis problemsfor complex nonlinear systems Among various kinds offuzzy methods Takagi-Sugeno (T-S) fuzzy models providea successful method to describe certain complex nonlinearsystem using some local linear subsystems (see [13ndash15] andreferences therein) As pointed out in the above relatedliterature the Markovian jumping T-S fuzzy mathematicalmodels have always found their extensive applications in

the real world However almost all the applications aregreatly dependent on the stability of systems which can oftencome down to the stability of the equilibrium solution forthe corresponding mathematical models So in this paperwe may consider the stability of the nonlinear 119901-Laplace(119901 gt 1) diffusion fuzzy equations with Markovian jumpingNote that when 119901 = 2 the so-called reaction-diffusionequations have been widely investigated (see [16ndash20] andreferences therein) For example under Dirichlet boundaryconditions the existence result of 120583-stability equilibriumsolution in the sense of 1198712 norm for a class of time-delayreaction-diffusion equations was obtained in [16] Motivatedby the abovementioned literature in this paper we willsynthetically employ Ekeland variational principle Sobolevimbedding theorem and the Lyapunov functional methodto study the existence of exponential 119901-stability nonconstantequilibrium solution in the sense of 119871119901 norm (119901 gt 1) fordelayed Markovian jumping fuzzy equations with nonlinear119901-Laplace diffusion (119901 gt 1)

Let 119903(119905) [0 +infin) rarr 119878 be a right-continuous Markovprocess on the complete probability space (S ϝP) with

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2015 Article ID 812150 10 pageshttpdxdoiorg1011552015812150

2 Advances in Mathematical Physics

a natural filtration ϝ119905119905⩾0 and take values in the finite space119878 ≜ 1 2 119873 with generatorprod = 120587119894119896 given by

P (119903 (119905 + 120575) = 119896 | 119903 (119905) = 119894)

=

120587119894119896120575 + 119900 (120575) 119896 = 119894

1 + 120587119894119896120575 + 119900 (120575) 119896 = 119894

(1)

where 120587119894119896 ⩾ 0 is transition probability rate from 119894 to 119896 (119896 = 119894)

and 120587119894119894 = minussum119896isin119878119896 =119894 120587119894119896 120575 gt 0 and lim120575rarr 0(119900(120575)120575) = 0Consider the following fuzzy T-S Markovian jumping 119901-

Laplace partial dynamic equationsFuzzy rule 119895 IF 1205961 is ]1198951 and 120596120581 is ]119895120581 THEN

119889119906119894 (119905 119909)

119889119905= D119894div (

1003816100381610038161003816nabla119906119894

1003816100381610038161003816119901minus2

nabla119906119894) minus 119887119894 (119906119894)

+ 119888119894119895 (119903 (119905)) 119891119894 (119906119894 (119905 119909))

+ 119889119894119895 (119903 (119905)) 119892119894 (119906119894 (119905 minus 120591119894 (119905) 119909)) + 119868119894

119894 isinN 119905 gt 0 119909 isin Ω

119906119894 (119905 119909) = 0 119894 isinN 119905 ⩾ 0 119909 isin 120597Ω

119906119894 (119904 119909) = 120601119894 (119904 119909)

minus 120591119894 (0) ⩽ 119904 ⩽ 0 0 ⩽ 120591119894 (119905) ⩽ 120591119894 119894 isinN 119905 ⩾ 0 119909 isin Ω

(2)

where 120596119896(119905) (119896 = 1 2 120581) is the premise variableand ]119895119896 (119895 = 1 2 119869 119896 = 1 2 120581) is the fuzzyset that is characterized by membership function 119869 isthe number of the IF-THEN rules and 120581 is the num-ber of the premise variables Denote the premise variablevector 120596(119905) = [1205961(119905) 1205962(119905) 120596120581(119905)] and 120588119895(120596(119905)) =

119908119895(120596(119905))sum119869

119896=1 119908119896(120596(119905)) where 119908119895(120596(119905)) 119877120581rarr [0 1] (119895 =

1 2 119869) is the membership function of the system withrespect to the fuzzy rule 119895 120588119895 can be regarded as thenormalized weight of each IF-THEN rule satisfying

120588119895 (120596 (119905)) ⩾ 0119869

sum

119895=1120588119895 (120596 (119905)) = 1 (3)

Using a singleton fuzzifier product fuzzy inference andweighted average defuzzifier system (2) is inferred as follows

119889119906119894 (119905 119909)

119889119905= D119894div (

1003816100381610038161003816nabla119906119894

1003816100381610038161003816119901minus2

nabla119906119894) minus 119887119894 (119906119894)

+

119869

sum

119895=1120588119895 (120596 (119905)) (119888119894119895 (119903 (119905)) 119891119894 (119906119894 (119905 119909))

+ 119889119894119895 (119903 (119905)) 119892119894 (119906119894 (119905 minus 120591119894 (119905) 119909))) + 119868119894

119894 isinN 119905 gt 0 119909 isin Ω

119906119894 (119905 119909) = 0 119894 isinN 119905 ⩾ 0 119909 isin 120597Ω

(4)

with the initial condition119906119894 (119904 119909) = 120601119894 (119904 119909)

minus 120591119894 (0) ⩽ 119904 ⩽ 0 0 ⩽ 120591119894 (119905) ⩽ 120591119894 119894 isinN 119905 ⩾ 0 119909 isin Ω(4

1015840)

where Ω is a bounded subset in 119877119898 with smooth boundary120597Ω 119901 gt 1 is a scalar andN = 1 2 119899 is a finite set Thedivergence div(|nabla119906119894(119905 119909)|

119901minus2nabla119906119894(119905 119909)) = Δ119901119906119894(119905 119909) is the 119901-

Laplacian (see [21] for details) In mode 119903(119905) = 119903 we denote119888119894119895(119903(119905)) = 119888

119903

119894119895and 119889119894119895(119903(119905)) = 119889

119903

119894119895 which imply the connection

strengths of the 119894th neuron on the 119895th neuron at the systemmode 119903(119905) = 119903 respectively For any given 119894 isin ND119894 gt 0 is acorresponding constant dependent on 119894

Remark 1 Our methods employed in this paper are differ-ent from those of previous related literature For examplehomomorphic mapping theory was employed to obtain theexistence of equilibrium of ordinary differential equationsin [22] topological degree theory was used to obtain theexistence of equilibrium for fuzzy ordinary differential equa-tions in [23] and of equilibrium for reaction-diffusion partialdifferential equations in [17] In this paper Ekeland varia-tional principle is originally proposed to solve the existence ofnonconstant equilibrium for nonlinear diffusion equationsNote that the abovementioned constant equilibrium point119906lowast= (119906

lowast

1 119906lowast

119899)119879 can actually be regarded as the special

case of our nonconstant equilibrium point 119906 = 119906lowast(119909) =

(119906lowast

1 (119909) 119906lowast

119899(119909))

119879 with 119906lowast

119894(119909) equiv constant for 119894 = 1 2 119899

In addition our criterion about existence is only involvedin the activation functions while other more parametersneed be considered in the proof of the existence of constantequilibrium point in those previous literatures (see Remark 8below for details)

2 Preparation

Throughout this paper we assume the following

(A1) There exists a positive definition matrix 119861 = diag(11986111198612 119861119899) such that

inf119904isin119877

1198871015840

119894(119904) ⩾ 119861119894 gt 0 forall119894 isinN (5)

(A2) There exist positive definite matrices 119865 = diag(1198651 1198652 119865119899) and 119866 = diag(1198661 1198662 119866119899) such that

1003816100381610038161003816119891119894 (119904) minus 119891119894 (119905)1003816100381610038161003816 ⩽ 119865119894 |119904 minus 119905|

1003816100381610038161003816119892119894 (119904) minus 119892119894 (119905)1003816100381610038161003816 ⩽ 119866119894 |119904 minus 119905|

119904 119905 isin 119877 119894 isinN

(6)

Definition 2 119906 = 119906lowast(119909) = (119906

lowast

1 (119909) 119906lowast

2 (119909) 119906lowast

119899(119909))

119879 iscalled a nonconstant equilibrium solution of PDEs (4) if andonly if 119906 = 119906

lowast(119909) satisfies (4) In addition the nonconstant

equilibrium solution 119906lowast(119909) of PDEs (4) is called stochastically

global exponential 119901-stability about 119871119901 norm if there areconstants119872 gt 0 and 120575 gt 0 for every stochastic field solution119906(119905 119909) = (1199061(119905 119909) 1199062(119905 119909) 119906119899(119905 119909))

119879 of PDEs (4) suchthat

119864 (1003817100381710038171003817119906 (119905 119909) minus 119906

lowast1003817100381710038171003817119871119901) ⩽ 119872119890minus120575(119905minus1199050) (7)

where one denotes by 119906119871119901 = (sum119899

119894=1 119906119894119901

119871119901)1119901 the 119871119901 norm

and denotes 119906119894119871119901 = (intΩ|119906119894|119889119909)

1119901 Usually 1199050 = 0

Advances in Mathematical Physics 3

Definition 3 Let 119883 be a Banach space 120593 isin 1198621(119883 119877) and 119888 isin

119877 120593 satisfies the (PS)119888 condition if any sequence 119906119899 sub 119883such that

120593 (119906119899) 997888rarr 119888

1205931015840(119906119899) 997888rarr 0

(8)

has a convergent subsequence By theway the above sequence119906119899 with 120593(119906119899) rarr 119888 and 1205931015840

(119906119899) rarr 0 is called the (PS)119888sequence of 120593 for a given 119888 isin 119877

The following lemma originated from the famous Sobolevimbedding theorem

Lemma 4 Let Ω be a bounded subset in 119877119898 with smooth

boundary 120597Ω For 1 lt 119902 lt 119901 there exist the correspondingpositive constants 1198881 and 119888119902 such that for any 120585 isin 119882

11199010 (Ω)

intΩ

10038161003816100381610038161205851003816100381610038161003816 119889119909 ⩽ 1198881

10038171003817100381710038171205851003817100381710038171003817

(intΩ

10038161003816100381610038161205851003816100381610038161003816119902119889119909)

1119902⩽ 119888119902

10038171003817100381710038171205851003817100381710038171003817

(9)

where the Sobolev space11988211199010 (Ω) is the completion of 119862infin

0 (Ω)

with respect to the norm 120585 = (intΩ|nabla120585|

119901119889119909)

1119901

In 1979 Ekeland proposed the following famous Eke-land variational principle and its proof in [24] As is wellknown Ekeland variational principle has been the mostimportant result in nonlinear analysis and has been appliedto optimization theory control theory economic equilibriumtheory critical point theory dynamic systems and so forthIn this paper we also need the following Ekeland variationalprinciple

Lemma 5 (Ekeland variational principle [24 Theorem 1])Let119883 be a complete metric space and let 120593 119883 rarr (minusinfin +infin]

be a lower semicontinuous function bounded from below andnot identical to +infin Let 120576 gt 0 be given and let 120585 isin 119883 be suchthat

120593 (120585) ⩽ inf119883120593+ 120576 (10)

Then there exists V isin 119883 such that120593 (V) ⩽ 120593 (120585)

dist (120585 V) ⩽ 1(11)

and for each 119908 = V in119883

120593 (119908) gt 120593 (V) minus 120576 dist (V 119908) (12)

3 Main Result

Theorem 6 Let 119901 gt 1 Assume that there exists a positivescalar 119902 = 11990211199022 with 1 lt 119902 lt 119901 such that

lim|119904|rarr+infin

119887119894 (119904)

119904119902minus1= lim

|119904|rarr+infin

119891119894 (119904)

119904119902minus1= lim

|119904|rarr+infin

119892119894 (119904)

119904119902minus1= 0

= 119887119894 (0) forall119894 isinN

(13)

where 1199021 is an odd number and so is 1199022 Assume in addition|119868119894| lt 119872

If there exist a sequence of positive scalars ℎ119894(119903(119905)) (119903 isin

119878 119894 isinN) such that

minusA+119890120573120591

1 minus 119870B lt 0 (14)

where 120591 = max119894isinN120591119894 1205911015840119894 (119905) ⩽ 119870 lt 1 for all 119894 isin N A =

diag(1198861 1198862 119886119899)B = diag(1198871 1198872 119887119899) 119886119894 = min119903isin119878119886119894(119903)119887119894 = max119903isin119878119887119894(119903) and

119886119894 (119903) = ℎ119894 (119903 (119905))[

[

119901119861119894 minus120573minus119901119865119894(

119869

sum

119895=1

10038161003816100381610038161003816119888119903

119894119895

10038161003816100381610038161003816)

minus (119901minus 1) 119866119894(

119869

sum

119895=1

10038161003816100381610038161003816119889119903

119894119895

10038161003816100381610038161003816)]

]

119887119894 (119903) = ℎ119894 (119903 (119905)) 119866119894(

119869

sum

119895=1

10038161003816100381610038161003816119889119903

119894119895

10038161003816100381610038161003816)

119903 isin 119878

(15)

then there exists a nonconstant equilibrium solution for PDEs(4) which is stochastically global exponential 119901-stability about119871119901 norm

Proof The following proof may be divided into two big steps

Step 1 Firstly we need prove that there exists a nonconstantequilibrium solution for (4)

Consider the functional

120593119894 (120578) = intΩ

[

[

1119901D119894

1003816100381610038161003816nabla1205781003816100381610038161003816119901+B119894 (120578)

minus

119869

sum

119895=1120588119895 (120596 (119905)) ((119888

119903

119894119895F119894 (120578) + 119889

119903

119894119895G119894 (120578))) minus 119868119894120578

]

]

119889119909

(16)

where B119894(119904) = int119904

0 119887119894(120579)119889120579 F119894(119904) = int119904

0 119891119894(120579)119889120579 G119894(119904) =

int119904

0 119892119894(120579)119889120579 and 120578 = 120578(119905 119909) isin 11988211199010 (Ω) where the Sobolev

space 11988211199010 (Ω) is the completion of 119862infin

0 (Ω) with respect tothe norm 120578 = (int

Ω|nabla120578|

119901119889119909)

1119901

It is obvious that 120593119894 isin 1198621(119882

11199010 (Ω) 119877) for all 119894 isin

N If its critical point 119906lowast

119894(119909) exists then 119906lowast

(119909) = (119906lowast

1 (119909)

119906lowast

2 (119909) 119906lowast

119899(119909119899))

119879 must be a nonconstant equilibrium solu-tion of (4) So we only need to prove the existence of thecritical point of 120593119894 for all 119894 isinN

Next it follows from (13) that there exists a large enough119886119872 gt 0 such that

1003816100381610038161003816119891119894 (119904)1003816100381610038161003816 +1003816100381610038161003816119892119894 (119904)

1003816100381610038161003816 ⩽ |119904|119902minus1

|119904| gt 119886119872 (17)

Furthermore we can conclude by the continuity of 119891119894 and 119892119894

that1003816100381610038161003816119891119894 (119904)

1003816100381610038161003816 +1003816100381610038161003816119892119894 (119904)

1003816100381610038161003816 ⩽ 119887119872 forall |119904| ⩽ 119886119872 (18)


and hence

1003816100381610038161003816119891119894 (119904)1003816100381610038161003816 +1003816100381610038161003816119892119894 (119904)

1003816100381610038161003816 ⩽ 119887119872 + |119904|119902minus1

forall119904 isin 119877 (19)

where 119887119872 gt 0 is a constantFor 119904 ⩾ 0 we can derive by 119902 = 11990211199022 and the restrictive

conditions on 1199021 1199022

1003816100381610038161003816F119894 (119904)1003816100381610038161003816 ⩽ int

119904

0

1003816100381610038161003816119891119894 (119910)1003816100381610038161003816 119889119910 ⩽ 119887119872119904 +

1119902119904119902= 119887119872 |119904| +

1119902|119904|

119902 (20)

where 119887119872 gt 0 is a constantOn the other hand for 119904 lt 0 we can get by 119902 = 11990211199022 and

the restrictive conditions on 1199021 1199022

1003816100381610038161003816F119894 (119904)1003816100381610038161003816 ⩽ int

0

119904

1003816100381610038161003816119891119894 (119910)1003816100381610038161003816 119889119910 ⩽ int

0

119904

(119887119872 +10038161003816100381610038161199101003816100381610038161003816119902minus1) 119889119910

⩽ 119887119872 |119904| +1119902|119904|

119902

(21)

Hence

1003816100381610038161003816F119894 (119904)1003816100381610038161003816 ⩽ 119887119872 |119904| +

1119902|119904|

119902 forall119904 isin 119877 (22)

Similarly we can also deduce that

1003816100381610038161003816G119894 (119904)1003816100381610038161003816 ⩽ 119887119872 |119904| +

1119902|119904|

119902 forall119904 isin 119877 (23)

Denote |119888119894119895| = max119903isin119878|119888119903

119894119895| and |119889119894119895| = max119903isin119878|119889

119903

119894119895| then

|119888119894119895| ⩾ 0 and |119889119894119895| ⩾ 0 are constants independent of 119903Similarly we can prove that there exist 1198861198941 gt 0 and 1198861198942 gt 0such that

119869

sum

119895=1120588119895 (120596 (119905)) (119888

119903

119894119895F119894 (119904) + 119889

119903

119894119895G119894 (119904))

⩽ (

119869

sum

119895=1

1003816100381610038161003816100381611988811989411989510038161003816100381610038161003816)1003816100381610038161003816F119894 (119904)

1003816100381610038161003816 +(

119869

sum

119895=1

10038161003816100381610038161003816119889119894119895

10038161003816100381610038161003816)1003816100381610038161003816G119894 (119904)

1003816100381610038161003816

⩽ 1198861198941 |119904| + 1198861198942 |119904|119902 forall119904 isin 119877

(24)

where 1198861198941 = 119887119872(sum119869

119895=1 |119888119894119895| + sum119869

119895=1 |119889119894119895|) and 1198861198942 =

(1119902)(sum119869

119895=1 |119888119894119895| + sum119869

119895=1 |119889119894119895|) are the constants independent of119903

Thereby we have

[

[

119869

sum

119895=1120588119895 (120596 (119905)) (119888

119903

119894119895F119894 (119904) + 119889

119903

119894119895G119894 (119904))

]

]

+ 119868119894119904

⩽ (1198861198941 +119872) |119904| + 1198861198942 |119904|119902 forall119904 isin 119877

(25)

Since 1198871015840119894(119904) gt 0 we know that 119887119894(119904) gt 119887119894(0) = 0 if 119904 gt 0 and

119887119894(119904) lt 119887119894(0) = 0 if 119904 lt 0 And hence B119894(119904) = int119904

0 119887119894(120579)119889120579 ⩾ 0for all 119904 isin 119877 Besides we know from the above analysisand the Sobolev imbedding theorem (Lemma 4) that there

exist positive constants 1198871198941 = 1198881(1198861198941 + 119872) and 1198871198942 = 119888119902

1199021198861198942

independent of 119903 such that

120593119894 (120578) ⩾1119901D119894

10038171003817100381710038171205781003817100381710038171003817119901minus (1198861198941 +119872)int

Ω

10038161003816100381610038161205781003816100381610038161003816 119889119909

minus 1198861198942 intΩ

10038161003816100381610038161205781003816100381610038161003816119902119889119909

⩾1119901D119894

10038171003817100381710038171205781003817100381710038171003817119901minus 1198871198942

10038171003817100381710038171205781003817100381710038171003817119902minus 1198871198941

10038171003817100381710038171205781003817100381710038171003817 119894 isinN

(26)

Denote120583(119904) = (1119901)D119894119904119901minus1198871198942119904

119902minus1198871198941119904Owing to119901 gt 119902 gt 1

there exists a large enough constant 1199040 gt 0 such that 120583(119904) gt 0for all |119904| gt 1199040 And hence

120583 (119904) ⩾ min0 min119904isin[minus1199040 1199040]

120583 (119904) 119904 isin 119877 (27)

which implies that 120593119894 is bounded below And the infimummay be defined as 119888119894 = inf119883120593119894 where we denote119883 = 119882

11199010 (Ω)

for convenience and denote by 119883lowast its dual space Define theoperators A119894B119894 119883 rarr 119883

lowast as follows

⟨A119894 (120578) 120585⟩ = D119894 intΩ

1003816100381610038161003816nabla1205781003816100381610038161003816119901minus2

nabla120578nabla120585 119889119909

⟨B119894 (120578) 120585⟩ = intΩ

(119887119894 (120578)

minus

119869

sum

119895=1120588119895 (120596 (119905)) (119888

119903

119894119895119891119894 (120578) + 119889

119903

119894119895119892119894 (120578)) minus 119868119894)120585119889119909

(28)

where 120578 120585 isin 119883 ⟨sdot sdot⟩ denotes the adjoint pair for 119883 and 119883lowastIt follows by (A1) and (A2) that all 119887119894 119891119894 119892119894 are continuousSimilarly to (19) we can get

1003816100381610038161003816119887119894 (119904)1003816100381610038161003816 +1003816100381610038161003816119891119894 (119904)

1003816100381610038161003816 +1003816100381610038161003816119892119894 (119904)

1003816100381610038161003816 ⩽ 119888119872 + |119904|119902minus1

forall119904 isin 119877 (29)

where 119888119872 is a constant we can conclude from (13) and (29)that there exist positive constants 1198861198943 and 1198861198944 independent of119903 such that

10038161003816100381610038161003816100381610038161003816100381610038161003816

119887119894 (119904) minus

119869

sum

119895=1120588119895 (120596 (119905)) (119888

119903

119894119895119891119894 (119904) + 119889

119903

119894119895119892119894 (119904)) minus 119868119894

10038161003816100381610038161003816100381610038161003816100381610038161003816

⩽ 119888119872 + |119904|119902minus1

+

119869

sum

119895=1

1003816100381610038161003816100381611988811989411989510038161003816100381610038161003816sdot1003816100381610038161003816119891119894 (119904)

1003816100381610038161003816 +(

119869

sum

119895=1

10038161003816100381610038161003816119889119903

119894119895

10038161003816100381610038161003816)

sdot1003816100381610038161003816119892119894 (119904)

1003816100381610038161003816 +119872 ⩽ 1198861198943 + 1198861198944 |119904|119902minus1

119904 isin 119877

(30)

Here 1198861198943 = 119872 + 119888119872 + 119888119872(sum119869

119895=1 |119888119894119895| + sum119869

119895=1 |119889119894119895|) and 1198861198944 =

1 + sum119869

119895=1 |119888119894119895| + sum119869

119895=1 |119889119894119895|So we know from [25 26] that both A119894 and Aminus1

119894are

continuous and B119894 is compactNext we claim that for each 119894 isinN the functional120593119894 must

satisfy the (PS)119888 condition if only every (PS)119888 sequence of 120593119894

is bounded


Indeed let 120578119899 be the (PS)119888 sequence of 120593119894 for any given119888 It is obvious that

⟨1205931015840

119894(120578119899) 120585⟩ = ⟨A119894 (120578119899) minusB119894 (120578119899) 120585⟩ forall120585 isin 119883 (31)

Owing to 1205931015840

119894(120578119899) rarr 0 we have (A119894(120578119899) minus B119894(120578119899)) rarr

0 If 120578119899 is bounded we know from the reflexivity of theBanach space1198821119901

0 (Ω) that there exists a weakly convergentsubsequence (say 120578119899) Since B119894 is compact B119894(120578119899) mustown a convergent subsequence which implies that A119894(120578119899)

must own a convergent subsequence And furthermorethe continuity of Aminus1

119894yields that 120578119899 owns a convergent

subsequenceBelow we only need to prove that every (PS)119888119894 sequence

(say 120578119899) of 120593119894 is boundedIndeed similarly to (26) we have

120593119894 (120578119899) ⩾1119901D119894

10038171003817100381710038171205781198991003817100381710038171003817119901minus 1198871198942

10038171003817100381710038171205781198991003817100381710038171003817119902minus 1198871198941

10038171003817100381710038171205781198991003817100381710038171003817 (32)

Owing to the boundedness of 120593119894(120578119899) it is not difficultto prove by the application of reduction to absurdity that 120578119899must be bounded in1198821119901

0 (Ω)Now we may define the metric for the space119883 as follows

dist (120578 120585) = 1003817100381710038171003817120578 minus 1205851003817100381710038171003817 (33)

Then 119883 is a complete metric space with the above metricFrom the continuity of 120593119894 and the above analysis we knowthat 120593 119882

11199010 (Ω) rarr (minusinfin +infin] is a lower semicontinuous

function and bounded from below Owing to 0 isin 11988211199010 (Ω)

we can compute and deduce that |120593119894(0)| ⩽ sum119869

119895=1(|119888119894119895||F119894(0)| +|119889119894119895||G119894(0)|) which implies 120593119894 equiv +infin forall119894 isinN

According to Ekeland variational principle for given 120576 =1119899 there exists 120578119899 isin 119883 such that

120593119894 (120578) gt 120593119894 (120578119899) minus1119899

1003817100381710038171003817120578 minus 1205781198991003817100381710038171003817 forall120578 = 120578119899 (34)

120593119894 (120578119899) lt 119888119894 +1119899 (35)

So we can deduce from (34)

100381710038171003817100381710038171205931015840

119894(120578119899)

10038171003817100381710038171003817= sup

120585=1

1003816100381610038161003816119889120593119894 (120578119899 120585)1003816100381610038161003816 ⩽

1119899 (36)

Here 119889120593119894(120578119899 120585) is Gateaux derivative of 120593119894 at 120578119899 and 1205931015840

119894(120578119899) is

Frechet derivative of 120593119894 at 120578119899 Besides (35) yields 120593119894(120578119899) rarr 119888119894

Thenwe can conclude from the PS119888119894 condition that there exista convergent subsequence 120578119899119895 sub 120578119899 and 119906

lowast

119894isin 119882

11199010 (Ω)

such that 120578119899119895 rarr 119906lowast

119894 Moreover the continuity of 120593119894 yields

120593119894(119906lowast

119894) = 119888119894 Hence 120593119894 has a critical point 119906

lowast

119894(119909) in119883 for all 119894 isin

N That is there exists an equilibrium 119906lowast= (119906

lowast

1 119906lowast

2 119906lowast

119899)119879

for (4)

Step 2 Below we will prove the exponential 119901-stability forthe equilibrium point 119906lowast

Consider the Lyapunov-Krasovskii functional as

119881119894 (119905 119903 (119905)) = 1198811119894 (119905 119903 (119905)) +1198812119894 (119905 119903 (119905)) 119894 isinN (37)

where

1198811119894 (119905 119903 (119905)) = 119890120573119905intΩ

ℎ119894 (119903 (119905))1003816100381610038161003816119906119894 (119905 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119909

1198812119894 (119905 119903 (119905))

=119887119894

1 minus 119870119890120573120591intΩ

int

119905

119905minus120591119894(119905)

119890120573119904 1003816100381610038161003816119906119894 (119904 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119904 119889119909

(38)

Let L be the weak infinitesimal operator then for any givenmode 119903(119905) = 119903 isin 119878 taking the derivative of 1198811119894(119905 119903) withrespect to 119905 along the trajectory of (4) yields

L1198811119894 (119905 119903)

= lim120575rarr 0+

1120575E [int

Ω

119890120573(119905+120575)

ℎ119894 (119903 (119905 + 120575))1003816100381610038161003816119906119894 (119905 + 120575 119909) minus 119906

lowast

119894

1003816100381610038161003816119901119889119909 |

119903 (119905) = 119903] minus 119890120573119905intΩ

ℎ119894 (119903 (119905))1003816100381610038161003816119906119894 (119905 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119909

= 119890120573119905

intΩ

ℎ119894 (119903 (119905)) 1199011003816100381610038161003816119906119894 minus119906

lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894)

sdot [

[

D119894

119898

sum

119896=1(120597

120597119909119896

(1003816100381610038161003816nabla119906119894

1003816100381610038161003816119901minus2 120597119906119894

120597119909119896

)minus120597

120597119909119896

(1003816100381610038161003816nabla119906

lowast

119894

1003816100381610038161003816119901minus2 120597119906

lowast

119894

120597119909119896

))

minus (119887119894 (119906119894) minus 119887119894 (119906lowast

119894)) +

119869

sum

119895=1120588119895 (120596 (119905)) 119888

119903

119894119895(119891119894 (119906119894) minus119891119894 (119906

lowast

119894))

+

119869

sum

119895=1120588119895 (120596 (119905)) 119889

119903

119894119895(119892119894 (119906119894 (119905 minus 120591119894 (119905) 119909)) minus 119892119894 (119906

lowast

119894))]

]

119889119909

+intΩ

sum

119896isin119878

120587119894119896ℎ119894 (119896)1003816100381610038161003816119906119894 (119905 119909) minus 119906

lowast

119894

1003816100381610038161003816119901119889119909+120573ℎ119894 (119903 (119905)) int

Ω

1003816100381610038161003816119906119894

minus119906lowast

119894

1003816100381610038161003816119901119889119909

(39)

Next we claim that

119898

sum

119896=1intΩ

D119894

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894) (

120597

120597119909119896

(1003816100381610038161003816nabla119906119894

1003816100381610038161003816119901minus2 120597119906119894

120597119909119896

)

minus120597

120597119909119896

(1003816100381610038161003816nabla119906

lowast

119894

1003816100381610038161003816119901minus2 120597119906

lowast

119894

120597119909119896

))119889119909 ⩽ 0

(40)

To verify (40) we have to prove firstly the followingproposition by the Yang inequality

Proposition 7 For 120578 120595 isin 11988211199010 (Ω) one has

(1003816100381610038161003816nabla120578

1003816100381610038161003816119901minus2

nabla120578minus1003816100381610038161003816nabla120595

1003816100381610038161003816119901minus2

nabla120595) (nabla120578 minusnabla120595) ⩾ 0 (41)


Proof In fact the Yang inequality yields

1003816100381610038161003816nabla1205781003816100381610038161003816119901minus2

nabla120578nabla120595 ⩽1003816100381610038161003816nabla120578

1003816100381610038161003816119901minus1 1003816100381610038161003816nabla120595

1003816100381610038161003816

⩽119901 minus 1119901

(1003816100381610038161003816nabla120578

1003816100381610038161003816119901minus1)119901(119901minus1)

+

1003816100381610038161003816nabla1205951003816100381610038161003816119901

119901

1003816100381610038161003816nabla1205951003816100381610038161003816119901minus2

nabla120595nabla120578 ⩽119901 minus 1119901

(1003816100381610038161003816nabla120595

1003816100381610038161003816119901minus1)119901(119901minus1)

+

1003816100381610038161003816nabla1205781003816100381610038161003816119901

119901

(42)

Synthesizing the above two inequalities results in (41)So we can get by Gauss formula the Dirichlet zero-

boundary value and Proposition 7

119898

sum

119896=1intΩ

D119894

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894)

sdot (120597

120597119909119896

(1003816100381610038161003816nabla119906119894

1003816100381610038161003816119901minus2 120597119906119894

120597119909119896

)minus120597

120597119909119896

(1003816100381610038161003816nabla119906

lowast

119894

1003816100381610038161003816119901minus2 120597119906

lowast

119894

120597119909119896

))119889119909

= minus

119898

sum

119896=1intΩ

D119894 (1003816100381610038161003816nabla119906119894

1003816100381610038161003816119901minus2 120597119906119894

120597119909119896

minus1003816100381610038161003816nabla119906

lowast

119894

1003816100381610038161003816119901minus2 120597119906

lowast

119894

120597119909119896

)

sdot120597

120597119909119896

(1003816100381610038161003816119906119894 minus 119906

lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus 119906lowast

119894)) 119889119909 = minusint

Ω

D119894 (119901

minus 1) 1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901minus2

[(1003816100381610038161003816nabla119906119894

1003816100381610038161003816119901minus2

nabla119906119894 minus1003816100381610038161003816nabla119906

lowast

119894

1003816100381610038161003816119901minus2

nabla119906lowast

119894)

sdot (nabla119906119894 minusnabla119906lowast

119894)] 119889119909 ⩽ 0

(43)

which proves (40)

In addition we get by (A1)

intΩ

ℎ119894 (119903 (119905)) 1199011003816100381610038161003816119906119894 minus119906

lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894)

sdot (119887119894 (119906119894) minus 119887119894 (119906lowast

119894)) 119889119909 ⩾ ℎ119894 (119903 (119905))

sdot 119901119861119894 intΩ

1003816100381610038161003816119906i minus119906lowast

119894

1003816100381610038161003816119901119889119909

(44)

Further we can derive by (A2)

intΩ

ℎ119894 (119903 (119905)) 1199011003816100381610038161003816119906119894 minus119906

lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894)

sdot

119869

sum

119895=1120588119895 (120596 (119905)) 119888

119903

119894119895(119891119894 (119906119894) minus119891119894 (119906

lowast

119894)) 119889119909 ⩽ ℎ119894 (119903 (119905))

sdot 119901119865119894(

119869

sum

119895=1

10038161003816100381610038161003816119888119903

119894119895

10038161003816100381610038161003816)int

Ω

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901119889119909

(45)

In addition (A2) and the Yang inequality yield

intΩ

ℎ119894 (119903 (119905)) 1199011003816100381610038161003816119906119894 minus119906

lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894)

sdot

119869

sum

119895=1120588119895 (120596 (119905)) 119889

119903

119894119895(119892119894 (119906119894 (119905 minus 120591119894 (119905) 119909)) minus 119892119894 (119906

lowast

119894))

⩽ ℎ119894 (119903 (119905)) 119866119894(

119869

sum

119895=1

10038161003816100381610038161003816119889119903

119894119895

10038161003816100381610038161003816) [(119901 minus 1) int

Ω

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901119889119909

+intΩ

1003816100381610038161003816119906119894 (119905 minus 120591119894 (119905) 119909) minus 119906lowast

119894

1003816100381610038161003816119901119889119909]

(46)

Combining the above analyses results in

L1198811119894 (119905 119903) ⩽ 119890120573119905[minus119886119894 (119903) int

Ω

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901119889119909

+ 119887119894 (119903) intΩ


119894

1003816100381610038161003816119901119889119909] forall119903 isin 119878

(47)

On the other hand we have

L1198812119894 ⩽119887119894

1 minus 119870119890120573120591intΩ

119890120573119905 1003816100381610038161003816119906119894 (119905 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119909

minus 119887119894 intΩ

119890120573119905 1003816100381610038161003816119906119894 (119905 minus 120591119894 (119905) 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119909

(48)

Thus we get by (14)

L119881119894 (119905 119903 (119905)) ⩽ 119890120573119905(minus119886119894 +

119887119894

1 minus 119870119890120573120591)int

Ω

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901119889119909

⩽ 0 forall119905 ⩾ 0(49)

So we can obtain by the Dynkin formula

119864119881 (119905 119903 (119905)) minus 119864119881 (0 119903 (0)) = 119864int119905

0L119881 (119904 119903 (119904)) 119889119904

⩽ 0 119905 ⩾ 0(50)

Hence we have

(min119903isin119878

ℎ119894 (119903 (119905))) 119864 (119890120573119905intΩ

1003816100381610038161003816119906119894 (119905 119909) minus 119906lowast

119894(119909)1003816100381610038161003816119901119889119909)

⩽ 119864119881 (119905 119903 (119905)) ⩽ 119864119881 (0 119903 (0))

⩽ 119864(max119903isin119878

ℎ119894 (119903 (119905)) intΩ

1003816100381610038161003816119906119894 (0 119909) minus 119906lowast

119894(119909)1003816100381610038161003816119901119889119909

+119887119894

1 minus 119870119890120573120591intΩ

int

0

minus120591

119890120573119904 1003816100381610038161003816119906119894 (119904 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119904 119889119909)

⩽ (max119903isin119878

ℎ119894 (119903 (119905)) +119887119894

1 minus 119870119890120573120591)

sdot supminus120591⩽119904⩽0

1198641003817100381710038171003817120601119894 (119904) minus 119906

lowast

119894

1003817100381710038171003817119901

119871119901

(51)


which implies

119864(intΩ

1003816100381610038161003816119906119894 (119905 119909) minus 119906lowast

119894(119909)1003816100381610038161003816119901119889119909)

⩽ ℎ119894119890minus120573119905 sup

minus120591⩽119904⩽01198641003817100381710038171003817120601119894 (119904) minus 119906

lowast

119894

1003817100381710038171003817119901

119871119901 119894 isinN

(52)

where ℎ119894 = (max119903isin119878ℎ119894(119903(119905))+(119887119894(1minus119870))119890120573120591)min119903isin119878ℎ119894(119903(119905)) gt

0 forall119894 isinNThen we have

1198641003817100381710038171003817119906 minus 119906

lowast1003817100381710038171003817119901

119871119901= sum

119894isinN

119864(intΩ

1003816100381610038161003816119906119894 (119905 119909) minus 119906lowast

119894(119909)1003816100381610038161003816119901119889119909)

⩽ (max119894isinN

ℎ119894) 119890minus1205731199051198641003817100381710038171003817120601 minus 119906

lowast1003817100381710038171003817119901

120591

(53)

where we denote 119864120601 minus 119906lowast119901

120591= sum119894isinN sup

minus120591⩽119904⩽0(119864120601119894(119904) minus

119906lowast

119894119901

119871119901)

Now we can conclude fromDefinition 2 that the noncon-stant equilibrium solution of (4) is stochastically exponen-tially 119901-stable about 119871119901 norm And that completes the proofof Theorem 6

Remark 8 In [27] existence theorems of stochastic differ-ential equations on 119905 isin [1199050 119879) were given under someconditions on activation functions where119879 gt 0 is a constantAnd in [28 29] existence theorems of stochastic differentialequations were presented under some conditions on function119881 isin 119862

12([1199050minus120591 119879)times119877

119899 119877+) Motivated by [27] we proposed

some conditions on activation functions to set up existencecriterion for the equilibrium solution of system (4) In [2223] the constant equilibrium solution 119906 = 119906

lowast for all 119905 isin

[1199050 +infin)was obtained by homomorphicmapping theory andmatrix theory or matrix theory and homotopy invariancetheorem where 119906lowast

= (119906lowast

1 119906lowast

2 119906lowast

119894 119906

lowast

119899) and each 119906lowast

119894

is a constant In this paper we also need to consider theequilibrium solution of (4) defined on [1199050 +infin) Differentfrom [22 23] we consider the nonconstant equilibriumsolution 119906 = 119906lowast

(119909) = (119906lowast

1 (119909) 119906lowast

2 (119909) 119906lowast

119894(119909) 119906

lowast

119899(119909)) for

all 119905 isin [1199050 +infin) This equilibrium solution is a solution fora nonlinear 119901-Laplacian elliptic partial differential equationwhose space frame may be considered as infinite dimensionfunction space1198821119901

0 (Ω) And variational method is always apowerful tool to solve the problem Although the variationalmethod is more complicated than homomorphic mappingmethod 119872-matrix method or homotopy invariance theo-rem our criterion about existence is only involved in theactivation functions (remark condition (14) is not usedin the proof of existence) and hence is simpler and moreeffective than other criteria such as 119872-matrix criteria andLMI-based criteria because LMI-based criteria or119872-matrixcriteria always involve the computerMATLAB programmingin practical application while our condition (13) is easy toverify So our existence criterion is actually simpler and moreeffective than LMI-based criteria and other criteria which isthe main contribution in this paper

Remark 9 LMI-based stability criteria or119872-matrix stabilitycriteria are always proposed in many literatures related to

the mean square stability (see eg [30ndash33] and referencestherein) However when 119901 gt 1 and 119901 = 2 119901-stability criteriaalways involve more complicated mathematical method andmathematical deduction For example the stability criteria in[34] are not simpler than our stability criterion inTheorem 6Similar phenomena exposed in many literatures related to 119901-stability (see [15 34ndash38]) Besides the nonlinear 119901-Laplacian(119901 gt 1) operator produces great difficulties in 119901-stabilityproof However our condition (14) is still a LMI conditionwhich can be computed and verified by computer MATLABLMI Toolbox in practical application

4 Numerical Example

Example 1 Consider the 5-Laplace fuzzy T-S dynamicalequations as follows

Fuzzy Ruler 1 IF 1205961(119905) is 1119890minus21205961(119905) THEN

1198891199061 (119905 119909)

119889119905

= D1div (1003816100381610038161003816nabla1199061

10038161003816100381610038163nabla1199061) minus 1198871 (1199061)

+ (119888119903

111198911 (1199061 (119905 119909)) + 119889119903

111198921 (1199061 (119905 minus 1205911 (119905) 119909)))

+ 1198681 119905 gt 0 119909 isin Ω

1198891199062 (119905 119909)

119889119905

= D2div (1003816100381610038161003816nabla1199062

10038161003816100381610038163nabla1199062) minus 1198872 (1199062)

+ (119888119903

211198912 (1199062 (119905 119909)) + 119889119903

211198922 (1199062 (119905 minus 1205912 (119905) 119909)))

+ 1198682 119905 gt 0 119909 isin Ω

119906119894 (119905 119909) = 0 119894 isinN 119905 ⩾ 0 119909 isin 120597Ω

(54a)

Fuzzy Ruler 2 IF 1205962(119905) is 1 minus 1119890minus21205961(119905) THEN

1198891199061 (119905 119909)

119889119905

= D1div (1003816100381610038161003816nabla1199061

10038161003816100381610038163nabla1199061) minus 1198871 (1199061)

+ (119888119903

121198911 (1199061 (119905 119909)) + 119889119903

121198921 (1199061 (119905 minus 1205911 (119905) 119909)))

+ 1198681 119905 gt 0 119909 isin Ω

1198891199062 (119905 119909)

119889119905

= D2div (1003816100381610038161003816nabla1199062

10038161003816100381610038163nabla1199062) minus 1198872 (1199062)

+ (119888119903

221198912 (1199062 (119905 119909)) + 119889119903

221198922 (1199062 (119905 minus 1205912 (119905) 119909)))

+ 1198682 119905 gt 0 119909 isin Ω

119906119894 (119905 119909) = 0 119894 isinN 119905 ⩾ 0 119909 isin 120597Ω

(54b)


0 5 10 15 20 25 300

0005

001

0015

002

0025Sectional curve of the state variable u1(t x)

u1(t 0261)

0 5 10 15 20 25 300

02

04

06

08

10

0005

001

0015

002

0025

Time t

Computer simulation of the state u1(t x)

Spacex

u1

20 25 30

02

04

6

8

Figure 1 The state variable 1199061(119905 119909)

equipped with the initial value

120601 (119904 119909) = (02 cos (2120587119909 (1 minus 119909))2 sin (120587119904)2

025 sin (6119909 (1 minus 119909))2 cos (3120587119904)2)

minus 83 le 119904 le 0

(55)

where 119909 isin Ω ≜ (1199091 1199092)119879isin 119877

2 |119909119894| lt 1 119894 isin N119873 = 1 2

119903 isin 119878 ≜ 1 2 119888111 = 119888121 = 01 119888211 = 119888

221 = 011 119888112 = 119888

122 =

015 119888212 = 012 = 119888222 119889

111 = 119889

121 = 01 119889211 = 119889

221 = 011

119889112 = 119889

122 = 015 119889212 = 012 = 119889

222 1198871(1199061) = 119906

31 + (12)1199061

1198872(1199062) = 11990632 + (13)1199062 and 1198611 = 12 and 1198612 = 13 Consider

1198911(1199061) = 021199061 sin 1199061 = 1198921(1199061) 1198912(1199062) = 031199062 + 02 cos 1199062 =1198922(1199062) and 1198651 = 1198661 = 04 1198652 = 1198662 = 05 let 120591119894(119905) equiv 83 119894 isinN and 120591 = 83 119870 = 0 denote ℎ119894(119903) = ℎ

119903

119894 119903 isin 119878 Let ℎ11 =

1 ℎ12 = 2 ℎ21 = 3 ℎ22 = 4 and 120573 = 001 Denote 119886119894(119903) = 119886119903

119894

and 119887119894(119903) = 119887119903

119894 then we can compute by computer MATLAB

that 11988611 = 15900 11988612 = 49860 11988621 = 10633 11988622 = 2486711988711 = 01000 11988712 = 02760 11988721 = 02500 and 11988722 = 04600 andhence 1198861 = 15900 1198862 = 10633 1198871 = 02760 1198872 = 04600 and

minus 1198861 +1198871

1 minus 119870119890120573120591= minus 09570 lt 0

minus 1198862 +1198872

1 minus 119870119890120573120591= minus 00084 lt 0

(56)

which imply that condition (14) is satisfied In additioncondition (13) is obviously satisfied Therefore there existsa nonconstant equilibrium solution for PDEs (54a)-(54b)which is stochastically global exponential119901-stability about119871119901

norm (see Figures 1 and 2)

5 Conclusions

The nonlinear 119901-Laplace (119901 gt 1 119901 = 2) brings great difficul-ties to the proof of the existence of the119901-stability nonconstantequilibrium solution for 119901-Laplace (119901 gt 1 119901 = 2) partialdifferential equations (PDEs) always need be considered inBanach space1198821119901

(Ω)while the common linear Laplace (119901 =2) PDEs can be studied in the setting of the special Hilbertspace 1198671 that can be orthogonally decomposed into thedirect sum of the eigenfunction spaces However by applyingEkeland variational principle and the Yang inequality andconstructing the suitable Lyapunov functional we overcomethose difficulties to obtain the existence of exponential 119901-stability nonconstant equilibrium solution for Markovianjumping 119901-Laplace (119901 gt 1) partial differential equations(PDEs) (4) under Dirichlet boundary condition As far aswe know it is the first time to apply the Ekeland variationalprinciple to solve the above problem so that some methodsused in this paper are different from those of many previousrelated literatures (see Remark 1) In addition the obtainedexistence criterion is only involved in the activation functionsso that the criterion is simpler and easier than other existencecriteria to be verified in practical application Moreover anumerical example shows the effectiveness of the proposedmethods owing to the large allowable variation range of time-delay

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Theauthors are very thankful to the anonymous reviewers fortheir suggestions which improved the quality of this paper


0 1 2 3 4 5 6 7 80

0002

0004

0006

0008

001

0012

0014

0016

0018


u2(t 0258)

0 1 2 3 4 5 6 7 8002

0406

081

0

0005

001

0015

002

0025

003

Time t


Spacex

u2

5 6 7002

046


This work was supported by the National Basic ResearchProgram of China (2010CB732501) by the Scientific ResearchFund of Science TechnologyDepartment of Sichuan Province(2012JY010) and by Sichuan Educational Committee ScienceFoundation (08ZB002 12ZB349 and 14ZA0274)

References

[1] Y H Li Y S Mi and C Mu ldquoProperties of positive solutionsfor a nonlocal non-linear diffusion equation with nonlocalnonlinear boundary conditionrdquoActaMathematica Scientia vol34 no 3 pp 748ndash758 2014

[2] YWang C Mu and Z Xiang ldquoBlowup of solutions to a porousmedium equation with nonlocal boundary conditionrdquo AppliedMathematics andComputation vol 192 no 2 pp 579ndash585 2007

[3] V A Galaktionov ldquoOn asymptotic self-similar behaviour for aquasilinear heat equation single point blow-uprdquo SIAM Journalon Mathematical Analysis vol 26 no 3 pp 675ndash693 1995

[4] A A Samarskii S P Kurdyumov V A Galaktionov and AP Mikhailov Blow-Up in Problems for Quasilinear ParabolicEquations Nauka Moscow Russia 1987 Walter de GruyterBerlin Germany 1995

[5] R S Cantrell and C Cosner ldquoDiffusive logistic equations withindefinite weights population models in disrupted environ-ments IIrdquo SIAM Journal on Mathematical Analysis vol 22 no4 pp 1043ndash1064 1989

[6] Z Q Wu J N Zhao J X Yin and H L LiNonlinear DiffusionEquations World Scientific Publishing River Edge NJ USA2001

[7] P Qingfei Z Zifang and H Jingchang ldquoStability of thestochastic reaction-diffusion neural network with time-varyingdelays and P-laplacianrdquo Journal of Applied Mathematics vol2012 Article ID 405939 10 pages 2012

[8] R Rao S Zhong and X Wang ldquoStochastic stability criteriawith LMI conditions for Markovian jumping impulsive BAM

neural networkswithmode-dependent time-varying delays andnonlinear reaction-diffusionrdquo Communications in NonlinearScience and Numerical Simulation vol 19 no 1 pp 258ndash2732014

[9] X R Wang R F Rao and S M Zhong ldquoLMI approach tostability analysis of Cohen-Grossberg neural networks with p-Laplace diffusionrdquo Journal of Applied Mathematics vol 2012Article ID 523812 12 pages 2012

[10] R Sathy and P Balasubramaniam ldquoStability analysis of fuzzyMarkovian jumping Cohen-Grossberg BAM neural networkswith mixed time-varying delaysrdquoCommunications in NonlinearScience and Numerical Simulation vol 16 no 4 pp 2054ndash20642011

[11] H Y Liu Y Ou J Hu and T Liu ldquoDelay-dependent stabil-ity analysis for continuous-time BAM neural networks withMarkovian jumping parametersrdquo Neural Networks vol 23 no3 pp 315ndash321 2010

[12] X Liu and H Xi ldquoStability analysis for neutral delay Marko-vian jump systems with nonlinear perturbations and partiallyunknown transition ratesrdquo Advances in Mathematical Physicsvol 2013 Article ID 592483 20 pages 2013

[13] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[14] K Tanaka andM Sugeno ldquoStability analysis and design of fuzzycontrol systemsrdquo Fuzzy Sets and Systems vol 45 no 2 pp 135ndash156 1992

[15] S Long and D Xu ldquoGlobal exponential 119901-stability of stochasticnon-autonomous Takagi-Sugeno fuzzy cellular neural networkswith time-varying delays and impulsesrdquo Fuzzy Sets and Systemsvol 253 pp 82ndash100 2014

[16] Z Chen andD Zhao ldquoStabilization effect of diffusion in delayedneural networks systems with Dirichlet boundary conditionsrdquoJournal of the Franklin Institute vol 348 no 10 pp 2884ndash28972011


[17] L Wang and D Xu ldquoGlobal exponential stability of Hopfieldreaction-diffusion neural networks with time-varying delaysrdquoScience in China Series F Information Sciences vol 46 no 6pp 466ndash474 2003

[18] Y G Kao C HWang H R Karimi and R Bi ldquoGlobal stabilityof coupled Markovian switching reaction-diffusion systems onnetworksrdquoNonlinear Analysis Hybrid Systems vol 13 pp 61ndash732014

[19] C Xu and J J Wei ldquoOn stability of two degenerate reaction-diffusion systemsrdquo Journal of Mathematical Analysis and Appli-cations vol 390 no 1 pp 126ndash135 2012

[20] C H Zhou H Y Zhang H B Zhang and C Y Dang ldquoGlobalexponential stability of impulsive fuzzy Cohen-Grossberg neu-ral networks with mixed delays and reaction-diffusion termsrdquoNeurocomputing vol 91 pp 67ndash76 2012

[21] P Lindqvise ldquoOn the equation 119889119894V(|nabla119906|119901minus2) + 120582|119906|119901minus2119906 = 0rdquoProceedings of the American Mathematical Society vol 109 pp159ndash164 1990

[22] H Chen S M Zhong and J L Shao ldquoExponential stability cri-terion for interval neural networks with discrete and distributeddelaysrdquo Applied Mathematics and Computation vol 250 pp121ndash130 2015

[23] X H Zhang and K L Li ldquoIntegro-differential inequality andstability of BAM FCNNs with time delays in the leakage termsand distributed delaysrdquo Journal of Inequalities and Applicationsvol 2011 article 43 2011

[24] I Ekeland ldquoNonconvex minimization problemsrdquo Bulletin of theAmerican Mathematical Society vol 1 no 3 pp 443ndash474 1979

[25] P Drabek Solvability and Bifurcations of Nonlinear Equationsvol 265 of Pitman Research Notes in Mathematics SeriesLongman Harlow 1992

[26] P Drabek A Kufner and F NicolosiQuasilinear Elliptic Equa-tions with Degenerations and Singularities Walter De GruyterBerlin Germany 1997

[27] D Y Xu X H Wang and Z G Yang ldquoFurther resultson existence-uniqueness for stochastic functional differentialequationsrdquo Science China Mathematics vol 56 no 6 pp 1169ndash1180 2013

[28] D Y Xu B Li S J Long and L Y Teng ldquoMoment estimateand existence for solutions of stochastic functional differentialequationsrdquoNonlinear AnalysisTheory Methods amp Applicationsvol 108 pp 128ndash143 2014

[29] D Y Xu B Li S J Long and L Y Teng ldquoCorrigendum tolsquoMoment estimate and existence for solutions of stochastic func-tional differential equationsrsquo [Nonlinear Anal TMA 108 (2014)128ndash143]rdquo Nonlinear Analysis Theory Methods amp Applicationsvol 114 pp 128ndash143 2015

[30] H G Zhang Z S Wang and D R Liu ldquoA comprehensivereview of stability analysis of continuous-time recurrent neuralnetworksrdquo IEEE Transactions on Neural Networks and LearningSystems vol 25 no 7 pp 1229ndash1262 2014

[31] H Zhang and Y Wang ldquoStability analysis of Markovian jump-ing stochastic Cohen-Grossberg neural networks with mixedtime delaysrdquo IEEE Transactions on Neural Networks vol 19 no2 pp 366ndash370 2008

[32] J K Tian Y M Li J Z Zhao and S M Zhong ldquoDelay-dependent stochastic stability criteria for Markovian jumpingneural networks with mode-dependent time-varying delaysand partially known transition ratesrdquo Applied Mathematics andComputation vol 218 no 9 pp 5769ndash5781 2012

[33] S J Long and D Y Xu ldquoGlobal exponential stability of non-autonomous cellular neural networks with impulses and time-varying delaysrdquo Communications in Nonlinear Science andNumerical Simulation vol 18 no 6 pp 1463ndash1472 2013

[34] B Li and D Y Xu ldquoExponential p-stability of stochasticrecurrent neural networks with mixed delays and Markovianswitchingrdquo Neurocomputing vol 103 pp 239ndash246 2013

[35] L Hu Y Ren and T Xu ldquoP-Moment stability of solutions tostochastic differential equations driven byG-BrownianmotionrdquoApplied Mathematics and Computation vol 230 pp 231ndash2372014

[36] X H Wang Q Y Guo and D Y Xu ldquoExponential 119901-stabilityof impulsive stochastic Cohen-Grossberg neural networks withmixed delaysrdquo Mathematics and Computers in Simulation vol79 no 5 pp 1698ndash1710 2009

[37] D S Li X H Wang and D Y Xu ldquoExistence and global p-exponential stability of periodic solution for impulsive stochas-tic neural networks with delaysrdquo Nonlinear Analysis HybridSystems vol 6 no 3 pp 847ndash858 2012

[38] Z G Yang D Y Xu and L Xiang ldquoExponential p-stability ofimpulsive stochastic differential equations with delaysrdquo PhysicsLetters A vol 359 no 2 pp 129ndash137 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


a natural filtration ϝ119905119905⩾0 and take values in the finite space119878 ≜ 1 2 119873 with generatorprod = 120587119894119896 given by

P (119903 (119905 + 120575) = 119896 | 119903 (119905) = 119894)

=

120587119894119896120575 + 119900 (120575) 119896 = 119894

1 + 120587119894119896120575 + 119900 (120575) 119896 = 119894

(1)

where 120587119894119896 ⩾ 0 is transition probability rate from 119894 to 119896 (119896 = 119894)

and 120587119894119894 = minussum119896isin119878119896 =119894 120587119894119896 120575 gt 0 and lim120575rarr 0(119900(120575)120575) = 0Consider the following fuzzy T-S Markovian jumping 119901-

Laplace partial dynamic equationsFuzzy rule 119895 IF 1205961 is ]1198951 and 120596120581 is ]119895120581 THEN

119889119906119894 (119905 119909)

119889119905= D119894div (

1003816100381610038161003816nabla119906119894

1003816100381610038161003816119901minus2

nabla119906119894) minus 119887119894 (119906119894)

+ 119888119894119895 (119903 (119905)) 119891119894 (119906119894 (119905 119909))

+ 119889119894119895 (119903 (119905)) 119892119894 (119906119894 (119905 minus 120591119894 (119905) 119909)) + 119868119894

119894 isinN 119905 gt 0 119909 isin Ω

119906119894 (119905 119909) = 0 119894 isinN 119905 ⩾ 0 119909 isin 120597Ω

119906119894 (119904 119909) = 120601119894 (119904 119909)

minus 120591119894 (0) ⩽ 119904 ⩽ 0 0 ⩽ 120591119894 (119905) ⩽ 120591119894 119894 isinN 119905 ⩾ 0 119909 isin Ω

(2)

where 120596119896(119905) (119896 = 1 2 120581) is the premise variableand ]119895119896 (119895 = 1 2 119869 119896 = 1 2 120581) is the fuzzyset that is characterized by membership function 119869 isthe number of the IF-THEN rules and 120581 is the num-ber of the premise variables Denote the premise variablevector 120596(119905) = [1205961(119905) 1205962(119905) 120596120581(119905)] and 120588119895(120596(119905)) =

119908119895(120596(119905))sum119869

119896=1 119908119896(120596(119905)) where 119908119895(120596(119905)) 119877120581rarr [0 1] (119895 =

1 2 119869) is the membership function of the system withrespect to the fuzzy rule 119895 120588119895 can be regarded as thenormalized weight of each IF-THEN rule satisfying

120588119895 (120596 (119905)) ⩾ 0119869

sum

119895=1120588119895 (120596 (119905)) = 1 (3)

Using a singleton fuzzifier product fuzzy inference andweighted average defuzzifier system (2) is inferred as follows

119889119906119894 (119905 119909)

119889119905= D119894div (

1003816100381610038161003816nabla119906119894

1003816100381610038161003816119901minus2

nabla119906119894) minus 119887119894 (119906119894)

+

119869

sum

119895=1120588119895 (120596 (119905)) (119888119894119895 (119903 (119905)) 119891119894 (119906119894 (119905 119909))

+ 119889119894119895 (119903 (119905)) 119892119894 (119906119894 (119905 minus 120591119894 (119905) 119909))) + 119868119894

119894 isinN 119905 gt 0 119909 isin Ω

119906119894 (119905 119909) = 0 119894 isinN 119905 ⩾ 0 119909 isin 120597Ω

(4)

with the initial condition119906119894 (119904 119909) = 120601119894 (119904 119909)

minus 120591119894 (0) ⩽ 119904 ⩽ 0 0 ⩽ 120591119894 (119905) ⩽ 120591119894 119894 isinN 119905 ⩾ 0 119909 isin Ω(4

1015840)

where Ω is a bounded subset in 119877119898 with smooth boundary120597Ω 119901 gt 1 is a scalar andN = 1 2 119899 is a finite set Thedivergence div(|nabla119906119894(119905 119909)|

119901minus2nabla119906119894(119905 119909)) = Δ119901119906119894(119905 119909) is the 119901-

Laplacian (see [21] for details) In mode 119903(119905) = 119903 we denote119888119894119895(119903(119905)) = 119888

119903

119894119895and 119889119894119895(119903(119905)) = 119889

119903

119894119895 which imply the connection

strengths of the 119894th neuron on the 119895th neuron at the systemmode 119903(119905) = 119903 respectively For any given 119894 isin ND119894 gt 0 is acorresponding constant dependent on 119894

Remark 1 Our methods employed in this paper are differ-ent from those of previous related literature For examplehomomorphic mapping theory was employed to obtain theexistence of equilibrium of ordinary differential equationsin [22] topological degree theory was used to obtain theexistence of equilibrium for fuzzy ordinary differential equa-tions in [23] and of equilibrium for reaction-diffusion partialdifferential equations in [17] In this paper Ekeland varia-tional principle is originally proposed to solve the existence ofnonconstant equilibrium for nonlinear diffusion equationsNote that the abovementioned constant equilibrium point119906lowast= (119906

lowast

1 119906lowast

119899)119879 can actually be regarded as the special

case of our nonconstant equilibrium point 119906 = 119906lowast(119909) =

(119906lowast

1 (119909) 119906lowast

119899(119909))

119879 with 119906lowast

119894(119909) equiv constant for 119894 = 1 2 119899

In addition our criterion about existence is only involvedin the activation functions while other more parametersneed be considered in the proof of the existence of constantequilibrium point in those previous literatures (see Remark 8below for details)

2 Preparation

Throughout this paper we assume the following

(A1) There exists a positive definition matrix 119861 = diag(11986111198612 119861119899) such that

inf119904isin119877

1198871015840

119894(119904) ⩾ 119861119894 gt 0 forall119894 isinN (5)

(A2) There exist positive definite matrices 119865 = diag(1198651 1198652 119865119899) and 119866 = diag(1198661 1198662 119866119899) such that

1003816100381610038161003816119891119894 (119904) minus 119891119894 (119905)1003816100381610038161003816 ⩽ 119865119894 |119904 minus 119905|

1003816100381610038161003816119892119894 (119904) minus 119892119894 (119905)1003816100381610038161003816 ⩽ 119866119894 |119904 minus 119905|

119904 119905 isin 119877 119894 isinN

(6)

Definition 2 119906 = 119906lowast(119909) = (119906

lowast

1 (119909) 119906lowast

2 (119909) 119906lowast

119899(119909))

119879 iscalled a nonconstant equilibrium solution of PDEs (4) if andonly if 119906 = 119906

lowast(119909) satisfies (4) In addition the nonconstant

equilibrium solution 119906lowast(119909) of PDEs (4) is called stochastically

global exponential 119901-stability about 119871119901 norm if there areconstants119872 gt 0 and 120575 gt 0 for every stochastic field solution119906(119905 119909) = (1199061(119905 119909) 1199062(119905 119909) 119906119899(119905 119909))

119879 of PDEs (4) suchthat

119864 (1003817100381710038171003817119906 (119905 119909) minus 119906

lowast1003817100381710038171003817119871119901) ⩽ 119872119890minus120575(119905minus1199050) (7)

where one denotes by 119906119871119901 = (sum119899

119894=1 119906119894119901

119871119901)1119901 the 119871119901 norm

and denotes 119906119894119871119901 = (intΩ|119906119894|119889119909)

1119901 Usually 1199050 = 0




120593 (119906119899) 997888rarr 119888

1205931015840(119906119899) 997888rarr 0

(8)






11199010 (Ω)

intΩ

10038161003816100381610038161205851003816100381610038161003816 119889119909 ⩽ 1198881

10038171003817100381710038171205851003817100381710038171003817

(intΩ

10038161003816100381610038161205851003816100381610038161003816119902119889119909)

1119902⩽ 119888119902

10038171003817100381710038171205851003817100381710038171003817

(9)


0 (Ω)


119901119889119909)

1119901




120593 (120585) ⩽ inf119883120593+ 120576 (10)


dist (120585 V) ⩽ 1(11)


120593 (119908) gt 120593 (V) minus 120576 dist (V 119908) (12)

3 Main Result



119887119894 (119904)

119904119902minus1= lim

|119904|rarr+infin

119891119894 (119904)

119904119902minus1= lim

|119904|rarr+infin

119892119894 (119904)

119904119902minus1= 0

= 119887119894 (0) forall119894 isinN

(13)




minusA+119890120573120591

1 minus 119870B lt 0 (14)



119886119894 (119903) = ℎ119894 (119903 (119905))[

[

119901119861119894 minus120573minus119901119865119894(

119869

sum

119895=1

10038161003816100381610038161003816119888119903

119894119895

10038161003816100381610038161003816)

minus (119901minus 1) 119866119894(

119869

sum

119895=1

10038161003816100381610038161003816119889119903

119894119895

10038161003816100381610038161003816)]

]

119887119894 (119903) = ℎ119894 (119903 (119905)) 119866119894(

119869

sum

119895=1

10038161003816100381610038161003816119889119903

119894119895

10038161003816100381610038161003816)

119903 isin 119878

(15)





120593119894 (120578) = intΩ

[

[

1119901D119894

1003816100381610038161003816nabla1205781003816100381610038161003816119901+B119894 (120578)

minus

119869

sum

119895=1120588119895 (120596 (119905)) ((119888

119903

119894119895F119894 (120578) + 119889

119903

119894119895G119894 (120578))) minus 119868119894120578

]

]

119889119909

(16)

where B119894(119904) = int119904

0 119887119894(120579)119889120579 F119894(119904) = int119904

0 119891119894(120579)119889120579 G119894(119904) =

int119904




Ω|nabla120578|

119901119889119909)

1119901


11199010 (Ω) 119877) for all 119894 isin



(119909) = (119906lowast

1 (119909)

119906lowast

2 (119909) 119906lowast

119899(119909119899))



1003816100381610038161003816119891119894 (119904)1003816100381610038161003816 +1003816100381610038161003816119892119894 (119904)

1003816100381610038161003816 ⩽ |119904|119902minus1

|119904| gt 119886119872 (17)


that1003816100381610038161003816119891119894 (119904)

1003816100381610038161003816 +1003816100381610038161003816119892119894 (119904)

1003816100381610038161003816 ⩽ 119887119872 forall |119904| ⩽ 119886119872 (18)


and hence

1003816100381610038161003816119891119894 (119904)1003816100381610038161003816 +1003816100381610038161003816119892119894 (119904)

1003816100381610038161003816 ⩽ 119887119872 + |119904|119902minus1

forall119904 isin 119877 (19)



1003816100381610038161003816F119894 (119904)1003816100381610038161003816 ⩽ int

119904

0

1003816100381610038161003816119891119894 (119910)1003816100381610038161003816 119889119910 ⩽ 119887119872119904 +

1119902119904119902= 119887119872 |119904| +

1119902|119904|

119902 (20)



1003816100381610038161003816F119894 (119904)1003816100381610038161003816 ⩽ int

0

119904

1003816100381610038161003816119891119894 (119910)1003816100381610038161003816 119889119910 ⩽ int

0

119904

(119887119872 +10038161003816100381610038161199101003816100381610038161003816119902minus1) 119889119910

⩽ 119887119872 |119904| +1119902|119904|

119902

(21)

Hence

1003816100381610038161003816F119894 (119904)1003816100381610038161003816 ⩽ 119887119872 |119904| +

1119902|119904|

119902 forall119904 isin 119877 (22)


1003816100381610038161003816G119894 (119904)1003816100381610038161003816 ⩽ 119887119872 |119904| +

1119902|119904|

119902 forall119904 isin 119877 (23)

Denote |119888119894119895| = max119903isin119878|119888119903

119894119895| and |119889119894119895| = max119903isin119878|119889

119903

119894119895| then


119869

sum

119895=1120588119895 (120596 (119905)) (119888

119903

119894119895F119894 (119904) + 119889

119903

119894119895G119894 (119904))

⩽ (

119869

sum

119895=1

1003816100381610038161003816100381611988811989411989510038161003816100381610038161003816)1003816100381610038161003816F119894 (119904)

1003816100381610038161003816 +(

119869

sum

119895=1

10038161003816100381610038161003816119889119894119895

10038161003816100381610038161003816)1003816100381610038161003816G119894 (119904)

1003816100381610038161003816

⩽ 1198861198941 |119904| + 1198861198942 |119904|119902 forall119904 isin 119877

(24)

where 1198861198941 = 119887119872(sum119869

119895=1 |119888119894119895| + sum119869

119895=1 |119889119894119895|) and 1198861198942 =

(1119902)(sum119869

119895=1 |119888119894119895| + sum119869


Thereby we have

[

[

119869

sum

119895=1120588119895 (120596 (119905)) (119888

119903

119894119895F119894 (119904) + 119889

119903

119894119895G119894 (119904))

]

]

+ 119868119894119904

⩽ (1198861198941 +119872) |119904| + 1198861198942 |119904|119902 forall119904 isin 119877

(25)





1199021198861198942


120593119894 (120578) ⩾1119901D119894

10038171003817100381710038171205781003817100381710038171003817119901minus (1198861198941 +119872)int

Ω

10038161003816100381610038161205781003816100381610038161003816 119889119909

minus 1198861198942 intΩ

10038161003816100381610038161205781003816100381610038161003816119902119889119909

⩾1119901D119894

10038171003817100381710038171205781003817100381710038171003817119901minus 1198871198942

10038171003817100381710038171205781003817100381710038171003817119902minus 1198871198941

10038171003817100381710038171205781003817100381710038171003817 119894 isinN

(26)

Denote120583(119904) = (1119901)D119894119904119901minus1198871198942119904




120583 (119904) 119904 isin 119877 (27)


11199010 (Ω)


lowast as follows

⟨A119894 (120578) 120585⟩ = D119894 intΩ

1003816100381610038161003816nabla1205781003816100381610038161003816119901minus2

nabla120578nabla120585 119889119909

⟨B119894 (120578) 120585⟩ = intΩ

(119887119894 (120578)

minus

119869

sum

119895=1120588119895 (120596 (119905)) (119888

119903

119894119895119891119894 (120578) + 119889

119903

119894119895119892119894 (120578)) minus 119868119894)120585119889119909

(28)


1003816100381610038161003816119887119894 (119904)1003816100381610038161003816 +1003816100381610038161003816119891119894 (119904)

1003816100381610038161003816 +1003816100381610038161003816119892119894 (119904)

1003816100381610038161003816 ⩽ 119888119872 + |119904|119902minus1

forall119904 isin 119877 (29)


10038161003816100381610038161003816100381610038161003816100381610038161003816

119887119894 (119904) minus

119869

sum

119895=1120588119895 (120596 (119905)) (119888

119903

119894119895119891119894 (119904) + 119889

119903

119894119895119892119894 (119904)) minus 119868119894

10038161003816100381610038161003816100381610038161003816100381610038161003816

⩽ 119888119872 + |119904|119902minus1

+

119869

sum

119895=1

1003816100381610038161003816100381611988811989411989510038161003816100381610038161003816sdot1003816100381610038161003816119891119894 (119904)

1003816100381610038161003816 +(

119869

sum

119895=1

10038161003816100381610038161003816119889119903

119894119895

10038161003816100381610038161003816)

sdot1003816100381610038161003816119892119894 (119904)

1003816100381610038161003816 +119872 ⩽ 1198861198943 + 1198861198944 |119904|119902minus1

119904 isin 119877

(30)

Here 1198861198943 = 119872 + 119888119872 + 119888119872(sum119869

119895=1 |119888119894119895| + sum119869

119895=1 |119889119894119895|) and 1198861198944 =

1 + sum119869

119895=1 |119888119894119895| + sum119869


119894are



is bounded



⟨1205931015840


Owing to 1205931015840








120593119894 (120578119899) ⩾1119901D119894

10038171003817100381710038171205781198991003817100381710038171003817119901minus 1198871198942

10038171003817100381710038171205781198991003817100381710038171003817119902minus 1198871198941

10038171003817100381710038171205781198991003817100381710038171003817 (32)



dist (120578 120585) = 1003817100381710038171003817120578 minus 1205851003817100381710038171003817 (33)







120593119894 (120578) gt 120593119894 (120578119899) minus1119899

1003817100381710038171003817120578 minus 1205781198991003817100381710038171003817 forall120578 = 120578119899 (34)

120593119894 (120578119899) lt 119888119894 +1119899 (35)


100381710038171003817100381710038171205931015840

119894(120578119899)

10038171003817100381710038171003817= sup

120585=1

1003816100381610038161003816119889120593119894 (120578119899 120585)1003816100381610038161003816 ⩽

1119899 (36)


119894(120578119899) is



lowast

119894isin 119882

11199010 (Ω)



120593119894(119906lowast


lowast

119894(119909) in119883 for all 119894 isin


lowast

1 119906lowast

2 119906lowast

119899)119879

for (4)



119881119894 (119905 119903 (119905)) = 1198811119894 (119905 119903 (119905)) +1198812119894 (119905 119903 (119905)) 119894 isinN (37)

where

1198811119894 (119905 119903 (119905)) = 119890120573119905intΩ

ℎ119894 (119903 (119905))1003816100381610038161003816119906119894 (119905 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119909

1198812119894 (119905 119903 (119905))

=119887119894

1 minus 119870119890120573120591intΩ

int

119905

119905minus120591119894(119905)

119890120573119904 1003816100381610038161003816119906119894 (119904 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119904 119889119909

(38)


L1198811119894 (119905 119903)

= lim120575rarr 0+

1120575E [int

Ω

119890120573(119905+120575)

ℎ119894 (119903 (119905 + 120575))1003816100381610038161003816119906119894 (119905 + 120575 119909) minus 119906

lowast

119894

1003816100381610038161003816119901119889119909 |

119903 (119905) = 119903] minus 119890120573119905intΩ

ℎ119894 (119903 (119905))1003816100381610038161003816119906119894 (119905 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119909

= 119890120573119905

intΩ

ℎ119894 (119903 (119905)) 1199011003816100381610038161003816119906119894 minus119906

lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894)

sdot [

[

D119894

119898

sum

119896=1(120597

120597119909119896

(1003816100381610038161003816nabla119906119894

1003816100381610038161003816119901minus2 120597119906119894

120597119909119896

)minus120597

120597119909119896

(1003816100381610038161003816nabla119906

lowast

119894

1003816100381610038161003816119901minus2 120597119906

lowast

119894

120597119909119896

))


119894)) +

119869

sum

119895=1120588119895 (120596 (119905)) 119888

119903

119894119895(119891119894 (119906119894) minus119891119894 (119906

lowast

119894))

+

119869

sum

119895=1120588119895 (120596 (119905)) 119889

119903

119894119895(119892119894 (119906119894 (119905 minus 120591119894 (119905) 119909)) minus 119892119894 (119906

lowast

119894))]

]

119889119909

+intΩ

sum

119896isin119878

120587119894119896ℎ119894 (119896)1003816100381610038161003816119906119894 (119905 119909) minus 119906

lowast

119894

1003816100381610038161003816119901119889119909+120573ℎ119894 (119903 (119905)) int

Ω

1003816100381610038161003816119906119894

minus119906lowast

119894

1003816100381610038161003816119901119889119909

(39)

Next we claim that

119898

sum

119896=1intΩ

D119894

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894) (

120597

120597119909119896

(1003816100381610038161003816nabla119906119894

1003816100381610038161003816119901minus2 120597119906119894

120597119909119896

)

minus120597

120597119909119896

(1003816100381610038161003816nabla119906

lowast

119894

1003816100381610038161003816119901minus2 120597119906

lowast

119894

120597119909119896

))119889119909 ⩽ 0

(40)



(1003816100381610038161003816nabla120578

1003816100381610038161003816119901minus2


1003816100381610038161003816119901minus2




1003816100381610038161003816nabla1205781003816100381610038161003816119901minus2


1003816100381610038161003816119901minus1 1003816100381610038161003816nabla120595

1003816100381610038161003816

⩽119901 minus 1119901

(1003816100381610038161003816nabla120578

1003816100381610038161003816119901minus1)119901(119901minus1)

+

1003816100381610038161003816nabla1205951003816100381610038161003816119901

119901

1003816100381610038161003816nabla1205951003816100381610038161003816119901minus2


(1003816100381610038161003816nabla120595

1003816100381610038161003816119901minus1)119901(119901minus1)

+

1003816100381610038161003816nabla1205781003816100381610038161003816119901

119901

(42)



119898

sum

119896=1intΩ

D119894

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894)

sdot (120597

120597119909119896

(1003816100381610038161003816nabla119906119894

1003816100381610038161003816119901minus2 120597119906119894

120597119909119896

)minus120597

120597119909119896

(1003816100381610038161003816nabla119906

lowast

119894

1003816100381610038161003816119901minus2 120597119906

lowast

119894

120597119909119896

))119889119909

= minus

119898

sum

119896=1intΩ

D119894 (1003816100381610038161003816nabla119906119894

1003816100381610038161003816119901minus2 120597119906119894

120597119909119896

minus1003816100381610038161003816nabla119906

lowast

119894

1003816100381610038161003816119901minus2 120597119906

lowast

119894

120597119909119896

)

sdot120597

120597119909119896

(1003816100381610038161003816119906119894 minus 119906

lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus 119906lowast

119894)) 119889119909 = minusint

Ω

D119894 (119901


119894

1003816100381610038161003816119901minus2

[(1003816100381610038161003816nabla119906119894

1003816100381610038161003816119901minus2


lowast

119894

1003816100381610038161003816119901minus2

nabla119906lowast

119894)


119894)] 119889119909 ⩽ 0

(43)

which proves (40)


intΩ

ℎ119894 (119903 (119905)) 1199011003816100381610038161003816119906119894 minus119906

lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894)


119894)) 119889119909 ⩾ ℎ119894 (119903 (119905))

sdot 119901119861119894 intΩ

1003816100381610038161003816119906i minus119906lowast

119894

1003816100381610038161003816119901119889119909

(44)


intΩ

ℎ119894 (119903 (119905)) 1199011003816100381610038161003816119906119894 minus119906

lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894)

sdot

119869

sum

119895=1120588119895 (120596 (119905)) 119888

119903

119894119895(119891119894 (119906119894) minus119891119894 (119906

lowast

119894)) 119889119909 ⩽ ℎ119894 (119903 (119905))

sdot 119901119865119894(

119869

sum

119895=1

10038161003816100381610038161003816119888119903

119894119895

10038161003816100381610038161003816)int

Ω

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901119889119909

(45)


intΩ

ℎ119894 (119903 (119905)) 1199011003816100381610038161003816119906119894 minus119906

lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894)

sdot

119869

sum

119895=1120588119895 (120596 (119905)) 119889

119903

119894119895(119892119894 (119906119894 (119905 minus 120591119894 (119905) 119909)) minus 119892119894 (119906

lowast

119894))

⩽ ℎ119894 (119903 (119905)) 119866119894(

119869

sum

119895=1

10038161003816100381610038161003816119889119903

119894119895

10038161003816100381610038161003816) [(119901 minus 1) int

Ω

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901119889119909

+intΩ


119894

1003816100381610038161003816119901119889119909]

(46)


L1198811119894 (119905 119903) ⩽ 119890120573119905[minus119886119894 (119903) int

Ω

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901119889119909

+ 119887119894 (119903) intΩ


119894

1003816100381610038161003816119901119889119909] forall119903 isin 119878

(47)


L1198812119894 ⩽119887119894

1 minus 119870119890120573120591intΩ

119890120573119905 1003816100381610038161003816119906119894 (119905 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119909

minus 119887119894 intΩ

119890120573119905 1003816100381610038161003816119906119894 (119905 minus 120591119894 (119905) 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119909

(48)

Thus we get by (14)

L119881119894 (119905 119903 (119905)) ⩽ 119890120573119905(minus119886119894 +

119887119894

1 minus 119870119890120573120591)int

Ω

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901119889119909

⩽ 0 forall119905 ⩾ 0(49)


119864119881 (119905 119903 (119905)) minus 119864119881 (0 119903 (0)) = 119864int119905

0L119881 (119904 119903 (119904)) 119889119904

⩽ 0 119905 ⩾ 0(50)

Hence we have

(min119903isin119878

ℎ119894 (119903 (119905))) 119864 (119890120573119905intΩ

1003816100381610038161003816119906119894 (119905 119909) minus 119906lowast

119894(119909)1003816100381610038161003816119901119889119909)

⩽ 119864119881 (119905 119903 (119905)) ⩽ 119864119881 (0 119903 (0))

⩽ 119864(max119903isin119878

ℎ119894 (119903 (119905)) intΩ

1003816100381610038161003816119906119894 (0 119909) minus 119906lowast

119894(119909)1003816100381610038161003816119901119889119909

+119887119894

1 minus 119870119890120573120591intΩ

int

0

minus120591

119890120573119904 1003816100381610038161003816119906119894 (119904 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119904 119889119909)

⩽ (max119903isin119878

ℎ119894 (119903 (119905)) +119887119894

1 minus 119870119890120573120591)


1198641003817100381710038171003817120601119894 (119904) minus 119906

lowast

119894

1003817100381710038171003817119901

119871119901

(51)


which implies

119864(intΩ

1003816100381610038161003816119906119894 (119905 119909) minus 119906lowast

119894(119909)1003816100381610038161003816119901119889119909)

⩽ ℎ119894119890minus120573119905 sup

minus120591⩽119904⩽01198641003817100381710038171003817120601119894 (119904) minus 119906

lowast

119894

1003817100381710038171003817119901

119871119901 119894 isinN

(52)



1198641003817100381710038171003817119906 minus 119906

lowast1003817100381710038171003817119901

119871119901= sum

119894isinN

119864(intΩ

1003816100381610038161003816119906119894 (119905 119909) minus 119906lowast

119894(119909)1003816100381610038161003816119901119889119909)

⩽ (max119894isinN

ℎ119894) 119890minus1205731199051198641003817100381710038171003817120601 minus 119906

lowast1003817100381710038171003817119901

120591

(53)



minus120591⩽119904⩽0(119864120601119894(119904) minus

119906lowast

119894119901

119871119901)



12([1199050minus120591 119879)times119877





= (119906lowast

1 119906lowast

2 119906lowast

119894 119906

lowast


119894


(119909) = (119906lowast

1 (119909) 119906lowast

2 (119909) 119906lowast

119894(119909) 119906

lowast

119899(119909)) for





4 Numerical Example



1198891199061 (119905 119909)

119889119905

= D1div (1003816100381610038161003816nabla1199061

10038161003816100381610038163nabla1199061) minus 1198871 (1199061)

+ (119888119903

111198911 (1199061 (119905 119909)) + 119889119903

111198921 (1199061 (119905 minus 1205911 (119905) 119909)))

+ 1198681 119905 gt 0 119909 isin Ω

1198891199062 (119905 119909)

119889119905

= D2div (1003816100381610038161003816nabla1199062

10038161003816100381610038163nabla1199062) minus 1198872 (1199062)

+ (119888119903

211198912 (1199062 (119905 119909)) + 119889119903

211198922 (1199062 (119905 minus 1205912 (119905) 119909)))

+ 1198682 119905 gt 0 119909 isin Ω

119906119894 (119905 119909) = 0 119894 isinN 119905 ⩾ 0 119909 isin 120597Ω

(54a)


1198891199061 (119905 119909)

119889119905

= D1div (1003816100381610038161003816nabla1199061

10038161003816100381610038163nabla1199061) minus 1198871 (1199061)

+ (119888119903

121198911 (1199061 (119905 119909)) + 119889119903

121198921 (1199061 (119905 minus 1205911 (119905) 119909)))

+ 1198681 119905 gt 0 119909 isin Ω

1198891199062 (119905 119909)

119889119905

= D2div (1003816100381610038161003816nabla1199062

10038161003816100381610038163nabla1199062) minus 1198872 (1199062)

+ (119888119903

221198912 (1199062 (119905 119909)) + 119889119903

221198922 (1199062 (119905 minus 1205912 (119905) 119909)))

+ 1198682 119905 gt 0 119909 isin Ω

119906119894 (119905 119909) = 0 119894 isinN 119905 ⩾ 0 119909 isin 120597Ω

(54b)


0 5 10 15 20 25 300

0005

001

0015

002


u1(t 0261)

0 5 10 15 20 25 300

02

04

06

08

10

0005

001

0015

002

0025

Time t


Spacex

u1

20 25 30

02

04

6

8



120601 (119904 119909) = (02 cos (2120587119909 (1 minus 119909))2 sin (120587119904)2

025 sin (6119909 (1 minus 119909))2 cos (3120587119904)2)

minus 83 le 119904 le 0

(55)

where 119909 isin Ω ≜ (1199091 1199092)119879isin 119877

2 |119909119894| lt 1 119894 isin N119873 = 1 2

119903 isin 119878 ≜ 1 2 119888111 = 119888121 = 01 119888211 = 119888

221 = 011 119888112 = 119888

122 =

015 119888212 = 012 = 119888222 119889

111 = 119889

121 = 01 119889211 = 119889

221 = 011

119889112 = 119889

122 = 015 119889212 = 012 = 119889

222 1198871(1199061) = 119906

31 + (12)1199061

1198872(1199062) = 11990632 + (13)1199062 and 1198611 = 12 and 1198612 = 13 Consider


119903

119894 119903 isin 119878 Let ℎ11 =

1 ℎ12 = 2 ℎ21 = 3 ℎ22 = 4 and 120573 = 001 Denote 119886119894(119903) = 119886119903

119894

and 119887119894(119903) = 119887119903



minus 1198861 +1198871

1 minus 119870119890120573120591= minus 09570 lt 0

minus 1198862 +1198872

1 minus 119870119890120573120591= minus 00084 lt 0

(56)



5 Conclusions





Acknowledgments



0 1 2 3 4 5 6 7 80

0002

0004

0006

0008

001

0012

0014

0016

0018


u2(t 0258)

0 1 2 3 4 5 6 7 8002

0406

081

0

0005

001

0015

002

0025

003

Time t


Spacex

u2

5 6 7002

046



References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120593 (119906119899) 997888rarr 119888

1205931015840(119906119899) 997888rarr 0

(8)






11199010 (Ω)

intΩ

10038161003816100381610038161205851003816100381610038161003816 119889119909 ⩽ 1198881

10038171003817100381710038171205851003817100381710038171003817

(intΩ

10038161003816100381610038161205851003816100381610038161003816119902119889119909)

1119902⩽ 119888119902

10038171003817100381710038171205851003817100381710038171003817

(9)


0 (Ω)


119901119889119909)

1119901




120593 (120585) ⩽ inf119883120593+ 120576 (10)


dist (120585 V) ⩽ 1(11)


120593 (119908) gt 120593 (V) minus 120576 dist (V 119908) (12)

3 Main Result



119887119894 (119904)

119904119902minus1= lim

|119904|rarr+infin

119891119894 (119904)

119904119902minus1= lim

|119904|rarr+infin

119892119894 (119904)

119904119902minus1= 0

= 119887119894 (0) forall119894 isinN

(13)




minusA+119890120573120591

1 minus 119870B lt 0 (14)



119886119894 (119903) = ℎ119894 (119903 (119905))[

[

119901119861119894 minus120573minus119901119865119894(

119869

sum

119895=1

10038161003816100381610038161003816119888119903

119894119895

10038161003816100381610038161003816)

minus (119901minus 1) 119866119894(

119869

sum

119895=1

10038161003816100381610038161003816119889119903

119894119895

10038161003816100381610038161003816)]

]

119887119894 (119903) = ℎ119894 (119903 (119905)) 119866119894(

119869

sum

119895=1

10038161003816100381610038161003816119889119903

119894119895

10038161003816100381610038161003816)

119903 isin 119878

(15)





120593119894 (120578) = intΩ

[

[

1119901D119894

1003816100381610038161003816nabla1205781003816100381610038161003816119901+B119894 (120578)

minus

119869

sum

119895=1120588119895 (120596 (119905)) ((119888

119903

119894119895F119894 (120578) + 119889

119903

119894119895G119894 (120578))) minus 119868119894120578

]

]

119889119909

(16)

where B119894(119904) = int119904

0 119887119894(120579)119889120579 F119894(119904) = int119904

0 119891119894(120579)119889120579 G119894(119904) =

int119904




Ω|nabla120578|

119901119889119909)

1119901


11199010 (Ω) 119877) for all 119894 isin



(119909) = (119906lowast

1 (119909)

119906lowast

2 (119909) 119906lowast

119899(119909119899))



1003816100381610038161003816119891119894 (119904)1003816100381610038161003816 +1003816100381610038161003816119892119894 (119904)

1003816100381610038161003816 ⩽ |119904|119902minus1

|119904| gt 119886119872 (17)


that1003816100381610038161003816119891119894 (119904)

1003816100381610038161003816 +1003816100381610038161003816119892119894 (119904)

1003816100381610038161003816 ⩽ 119887119872 forall |119904| ⩽ 119886119872 (18)


and hence

1003816100381610038161003816119891119894 (119904)1003816100381610038161003816 +1003816100381610038161003816119892119894 (119904)

1003816100381610038161003816 ⩽ 119887119872 + |119904|119902minus1

forall119904 isin 119877 (19)



1003816100381610038161003816F119894 (119904)1003816100381610038161003816 ⩽ int

119904

0

1003816100381610038161003816119891119894 (119910)1003816100381610038161003816 119889119910 ⩽ 119887119872119904 +

1119902119904119902= 119887119872 |119904| +

1119902|119904|

119902 (20)



1003816100381610038161003816F119894 (119904)1003816100381610038161003816 ⩽ int

0

119904

1003816100381610038161003816119891119894 (119910)1003816100381610038161003816 119889119910 ⩽ int

0

119904

(119887119872 +10038161003816100381610038161199101003816100381610038161003816119902minus1) 119889119910

⩽ 119887119872 |119904| +1119902|119904|

119902

(21)

Hence

1003816100381610038161003816F119894 (119904)1003816100381610038161003816 ⩽ 119887119872 |119904| +

1119902|119904|

119902 forall119904 isin 119877 (22)


1003816100381610038161003816G119894 (119904)1003816100381610038161003816 ⩽ 119887119872 |119904| +

1119902|119904|

119902 forall119904 isin 119877 (23)

Denote |119888119894119895| = max119903isin119878|119888119903

119894119895| and |119889119894119895| = max119903isin119878|119889

119903

119894119895| then


119869

sum

119895=1120588119895 (120596 (119905)) (119888

119903

119894119895F119894 (119904) + 119889

119903

119894119895G119894 (119904))

⩽ (

119869

sum

119895=1

1003816100381610038161003816100381611988811989411989510038161003816100381610038161003816)1003816100381610038161003816F119894 (119904)

1003816100381610038161003816 +(

119869

sum

119895=1

10038161003816100381610038161003816119889119894119895

10038161003816100381610038161003816)1003816100381610038161003816G119894 (119904)

1003816100381610038161003816

⩽ 1198861198941 |119904| + 1198861198942 |119904|119902 forall119904 isin 119877

(24)

where 1198861198941 = 119887119872(sum119869

119895=1 |119888119894119895| + sum119869

119895=1 |119889119894119895|) and 1198861198942 =

(1119902)(sum119869

119895=1 |119888119894119895| + sum119869


Thereby we have

[

[

119869

sum

119895=1120588119895 (120596 (119905)) (119888

119903

119894119895F119894 (119904) + 119889

119903

119894119895G119894 (119904))

]

]

+ 119868119894119904

⩽ (1198861198941 +119872) |119904| + 1198861198942 |119904|119902 forall119904 isin 119877

(25)





1199021198861198942


120593119894 (120578) ⩾1119901D119894

10038171003817100381710038171205781003817100381710038171003817119901minus (1198861198941 +119872)int

Ω

10038161003816100381610038161205781003816100381610038161003816 119889119909

minus 1198861198942 intΩ

10038161003816100381610038161205781003816100381610038161003816119902119889119909

⩾1119901D119894

10038171003817100381710038171205781003817100381710038171003817119901minus 1198871198942

10038171003817100381710038171205781003817100381710038171003817119902minus 1198871198941

10038171003817100381710038171205781003817100381710038171003817 119894 isinN

(26)

Denote120583(119904) = (1119901)D119894119904119901minus1198871198942119904




120583 (119904) 119904 isin 119877 (27)


11199010 (Ω)


lowast as follows

⟨A119894 (120578) 120585⟩ = D119894 intΩ

1003816100381610038161003816nabla1205781003816100381610038161003816119901minus2

nabla120578nabla120585 119889119909

⟨B119894 (120578) 120585⟩ = intΩ

(119887119894 (120578)

minus

119869

sum

119895=1120588119895 (120596 (119905)) (119888

119903

119894119895119891119894 (120578) + 119889

119903

119894119895119892119894 (120578)) minus 119868119894)120585119889119909

(28)


1003816100381610038161003816119887119894 (119904)1003816100381610038161003816 +1003816100381610038161003816119891119894 (119904)

1003816100381610038161003816 +1003816100381610038161003816119892119894 (119904)

1003816100381610038161003816 ⩽ 119888119872 + |119904|119902minus1

forall119904 isin 119877 (29)


10038161003816100381610038161003816100381610038161003816100381610038161003816

119887119894 (119904) minus

119869

sum

119895=1120588119895 (120596 (119905)) (119888

119903

119894119895119891119894 (119904) + 119889

119903

119894119895119892119894 (119904)) minus 119868119894

10038161003816100381610038161003816100381610038161003816100381610038161003816

⩽ 119888119872 + |119904|119902minus1

+

119869

sum

119895=1

1003816100381610038161003816100381611988811989411989510038161003816100381610038161003816sdot1003816100381610038161003816119891119894 (119904)

1003816100381610038161003816 +(

119869

sum

119895=1

10038161003816100381610038161003816119889119903

119894119895

10038161003816100381610038161003816)

sdot1003816100381610038161003816119892119894 (119904)

1003816100381610038161003816 +119872 ⩽ 1198861198943 + 1198861198944 |119904|119902minus1

119904 isin 119877

(30)

Here 1198861198943 = 119872 + 119888119872 + 119888119872(sum119869

119895=1 |119888119894119895| + sum119869

119895=1 |119889119894119895|) and 1198861198944 =

1 + sum119869

119895=1 |119888119894119895| + sum119869


119894are



is bounded



⟨1205931015840


Owing to 1205931015840








120593119894 (120578119899) ⩾1119901D119894

10038171003817100381710038171205781198991003817100381710038171003817119901minus 1198871198942

10038171003817100381710038171205781198991003817100381710038171003817119902minus 1198871198941

10038171003817100381710038171205781198991003817100381710038171003817 (32)



dist (120578 120585) = 1003817100381710038171003817120578 minus 1205851003817100381710038171003817 (33)







120593119894 (120578) gt 120593119894 (120578119899) minus1119899

1003817100381710038171003817120578 minus 1205781198991003817100381710038171003817 forall120578 = 120578119899 (34)

120593119894 (120578119899) lt 119888119894 +1119899 (35)


100381710038171003817100381710038171205931015840

119894(120578119899)

10038171003817100381710038171003817= sup

120585=1

1003816100381610038161003816119889120593119894 (120578119899 120585)1003816100381610038161003816 ⩽

1119899 (36)


119894(120578119899) is



lowast

119894isin 119882

11199010 (Ω)



120593119894(119906lowast


lowast

119894(119909) in119883 for all 119894 isin


lowast

1 119906lowast

2 119906lowast

119899)119879

for (4)



119881119894 (119905 119903 (119905)) = 1198811119894 (119905 119903 (119905)) +1198812119894 (119905 119903 (119905)) 119894 isinN (37)

where

1198811119894 (119905 119903 (119905)) = 119890120573119905intΩ

ℎ119894 (119903 (119905))1003816100381610038161003816119906119894 (119905 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119909

1198812119894 (119905 119903 (119905))

=119887119894

1 minus 119870119890120573120591intΩ

int

119905

119905minus120591119894(119905)

119890120573119904 1003816100381610038161003816119906119894 (119904 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119904 119889119909

(38)


L1198811119894 (119905 119903)

= lim120575rarr 0+

1120575E [int

Ω

119890120573(119905+120575)

ℎ119894 (119903 (119905 + 120575))1003816100381610038161003816119906119894 (119905 + 120575 119909) minus 119906

lowast

119894

1003816100381610038161003816119901119889119909 |

119903 (119905) = 119903] minus 119890120573119905intΩ

ℎ119894 (119903 (119905))1003816100381610038161003816119906119894 (119905 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119909

= 119890120573119905

intΩ

ℎ119894 (119903 (119905)) 1199011003816100381610038161003816119906119894 minus119906

lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894)

sdot [

[

D119894

119898

sum

119896=1(120597

120597119909119896

(1003816100381610038161003816nabla119906119894

1003816100381610038161003816119901minus2 120597119906119894

120597119909119896

)minus120597

120597119909119896

(1003816100381610038161003816nabla119906

lowast

119894

1003816100381610038161003816119901minus2 120597119906

lowast

119894

120597119909119896

))


119894)) +

119869

sum

119895=1120588119895 (120596 (119905)) 119888

119903

119894119895(119891119894 (119906119894) minus119891119894 (119906

lowast

119894))

+

119869

sum

119895=1120588119895 (120596 (119905)) 119889

119903

119894119895(119892119894 (119906119894 (119905 minus 120591119894 (119905) 119909)) minus 119892119894 (119906

lowast

119894))]

]

119889119909

+intΩ

sum

119896isin119878

120587119894119896ℎ119894 (119896)1003816100381610038161003816119906119894 (119905 119909) minus 119906

lowast

119894

1003816100381610038161003816119901119889119909+120573ℎ119894 (119903 (119905)) int

Ω

1003816100381610038161003816119906119894

minus119906lowast

119894

1003816100381610038161003816119901119889119909

(39)

Next we claim that

119898

sum

119896=1intΩ

D119894

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894) (

120597

120597119909119896

(1003816100381610038161003816nabla119906119894

1003816100381610038161003816119901minus2 120597119906119894

120597119909119896

)

minus120597

120597119909119896

(1003816100381610038161003816nabla119906

lowast

119894

1003816100381610038161003816119901minus2 120597119906

lowast

119894

120597119909119896

))119889119909 ⩽ 0

(40)



(1003816100381610038161003816nabla120578

1003816100381610038161003816119901minus2


1003816100381610038161003816119901minus2




1003816100381610038161003816nabla1205781003816100381610038161003816119901minus2


1003816100381610038161003816119901minus1 1003816100381610038161003816nabla120595

1003816100381610038161003816

⩽119901 minus 1119901

(1003816100381610038161003816nabla120578

1003816100381610038161003816119901minus1)119901(119901minus1)

+

1003816100381610038161003816nabla1205951003816100381610038161003816119901

119901

1003816100381610038161003816nabla1205951003816100381610038161003816119901minus2


(1003816100381610038161003816nabla120595

1003816100381610038161003816119901minus1)119901(119901minus1)

+

1003816100381610038161003816nabla1205781003816100381610038161003816119901

119901

(42)



119898

sum

119896=1intΩ

D119894

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894)

sdot (120597

120597119909119896

(1003816100381610038161003816nabla119906119894

1003816100381610038161003816119901minus2 120597119906119894

120597119909119896

)minus120597

120597119909119896

(1003816100381610038161003816nabla119906

lowast

119894

1003816100381610038161003816119901minus2 120597119906

lowast

119894

120597119909119896

))119889119909

= minus

119898

sum

119896=1intΩ

D119894 (1003816100381610038161003816nabla119906119894

1003816100381610038161003816119901minus2 120597119906119894

120597119909119896

minus1003816100381610038161003816nabla119906

lowast

119894

1003816100381610038161003816119901minus2 120597119906

lowast

119894

120597119909119896

)

sdot120597

120597119909119896

(1003816100381610038161003816119906119894 minus 119906

lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus 119906lowast

119894)) 119889119909 = minusint

Ω

D119894 (119901


119894

1003816100381610038161003816119901minus2

[(1003816100381610038161003816nabla119906119894

1003816100381610038161003816119901minus2


lowast

119894

1003816100381610038161003816119901minus2

nabla119906lowast

119894)


119894)] 119889119909 ⩽ 0

(43)

which proves (40)


intΩ

ℎ119894 (119903 (119905)) 1199011003816100381610038161003816119906119894 minus119906

lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894)


119894)) 119889119909 ⩾ ℎ119894 (119903 (119905))

sdot 119901119861119894 intΩ

1003816100381610038161003816119906i minus119906lowast

119894

1003816100381610038161003816119901119889119909

(44)


intΩ

ℎ119894 (119903 (119905)) 1199011003816100381610038161003816119906119894 minus119906

lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894)

sdot

119869

sum

119895=1120588119895 (120596 (119905)) 119888

119903

119894119895(119891119894 (119906119894) minus119891119894 (119906

lowast

119894)) 119889119909 ⩽ ℎ119894 (119903 (119905))

sdot 119901119865119894(

119869

sum

119895=1

10038161003816100381610038161003816119888119903

119894119895

10038161003816100381610038161003816)int

Ω

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901119889119909

(45)


intΩ

ℎ119894 (119903 (119905)) 1199011003816100381610038161003816119906119894 minus119906

lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894)

sdot

119869

sum

119895=1120588119895 (120596 (119905)) 119889

119903

119894119895(119892119894 (119906119894 (119905 minus 120591119894 (119905) 119909)) minus 119892119894 (119906

lowast

119894))

⩽ ℎ119894 (119903 (119905)) 119866119894(

119869

sum

119895=1

10038161003816100381610038161003816119889119903

119894119895

10038161003816100381610038161003816) [(119901 minus 1) int

Ω

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901119889119909

+intΩ


119894

1003816100381610038161003816119901119889119909]

(46)


L1198811119894 (119905 119903) ⩽ 119890120573119905[minus119886119894 (119903) int

Ω

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901119889119909

+ 119887119894 (119903) intΩ


119894

1003816100381610038161003816119901119889119909] forall119903 isin 119878

(47)


L1198812119894 ⩽119887119894

1 minus 119870119890120573120591intΩ

119890120573119905 1003816100381610038161003816119906119894 (119905 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119909

minus 119887119894 intΩ

119890120573119905 1003816100381610038161003816119906119894 (119905 minus 120591119894 (119905) 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119909

(48)

Thus we get by (14)

L119881119894 (119905 119903 (119905)) ⩽ 119890120573119905(minus119886119894 +

119887119894

1 minus 119870119890120573120591)int

Ω

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901119889119909

⩽ 0 forall119905 ⩾ 0(49)


119864119881 (119905 119903 (119905)) minus 119864119881 (0 119903 (0)) = 119864int119905

0L119881 (119904 119903 (119904)) 119889119904

⩽ 0 119905 ⩾ 0(50)

Hence we have

(min119903isin119878

ℎ119894 (119903 (119905))) 119864 (119890120573119905intΩ

1003816100381610038161003816119906119894 (119905 119909) minus 119906lowast

119894(119909)1003816100381610038161003816119901119889119909)

⩽ 119864119881 (119905 119903 (119905)) ⩽ 119864119881 (0 119903 (0))

⩽ 119864(max119903isin119878

ℎ119894 (119903 (119905)) intΩ

1003816100381610038161003816119906119894 (0 119909) minus 119906lowast

119894(119909)1003816100381610038161003816119901119889119909

+119887119894

1 minus 119870119890120573120591intΩ

int

0

minus120591

119890120573119904 1003816100381610038161003816119906119894 (119904 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119904 119889119909)

⩽ (max119903isin119878

ℎ119894 (119903 (119905)) +119887119894

1 minus 119870119890120573120591)


1198641003817100381710038171003817120601119894 (119904) minus 119906

lowast

119894

1003817100381710038171003817119901

119871119901

(51)


which implies

119864(intΩ

1003816100381610038161003816119906119894 (119905 119909) minus 119906lowast

119894(119909)1003816100381610038161003816119901119889119909)

⩽ ℎ119894119890minus120573119905 sup

minus120591⩽119904⩽01198641003817100381710038171003817120601119894 (119904) minus 119906

lowast

119894

1003817100381710038171003817119901

119871119901 119894 isinN

(52)



1198641003817100381710038171003817119906 minus 119906

lowast1003817100381710038171003817119901

119871119901= sum

119894isinN

119864(intΩ

1003816100381610038161003816119906119894 (119905 119909) minus 119906lowast

119894(119909)1003816100381610038161003816119901119889119909)

⩽ (max119894isinN

ℎ119894) 119890minus1205731199051198641003817100381710038171003817120601 minus 119906

lowast1003817100381710038171003817119901

120591

(53)



minus120591⩽119904⩽0(119864120601119894(119904) minus

119906lowast

119894119901

119871119901)



12([1199050minus120591 119879)times119877





= (119906lowast

1 119906lowast

2 119906lowast

119894 119906

lowast


119894


(119909) = (119906lowast

1 (119909) 119906lowast

2 (119909) 119906lowast

119894(119909) 119906

lowast

119899(119909)) for





4 Numerical Example



1198891199061 (119905 119909)

119889119905

= D1div (1003816100381610038161003816nabla1199061

10038161003816100381610038163nabla1199061) minus 1198871 (1199061)

+ (119888119903

111198911 (1199061 (119905 119909)) + 119889119903

111198921 (1199061 (119905 minus 1205911 (119905) 119909)))

+ 1198681 119905 gt 0 119909 isin Ω

1198891199062 (119905 119909)

119889119905

= D2div (1003816100381610038161003816nabla1199062

10038161003816100381610038163nabla1199062) minus 1198872 (1199062)

+ (119888119903

211198912 (1199062 (119905 119909)) + 119889119903

211198922 (1199062 (119905 minus 1205912 (119905) 119909)))

+ 1198682 119905 gt 0 119909 isin Ω

119906119894 (119905 119909) = 0 119894 isinN 119905 ⩾ 0 119909 isin 120597Ω

(54a)


1198891199061 (119905 119909)

119889119905

= D1div (1003816100381610038161003816nabla1199061

10038161003816100381610038163nabla1199061) minus 1198871 (1199061)

+ (119888119903

121198911 (1199061 (119905 119909)) + 119889119903

121198921 (1199061 (119905 minus 1205911 (119905) 119909)))

+ 1198681 119905 gt 0 119909 isin Ω

1198891199062 (119905 119909)

119889119905

= D2div (1003816100381610038161003816nabla1199062

10038161003816100381610038163nabla1199062) minus 1198872 (1199062)

+ (119888119903

221198912 (1199062 (119905 119909)) + 119889119903

221198922 (1199062 (119905 minus 1205912 (119905) 119909)))

+ 1198682 119905 gt 0 119909 isin Ω

119906119894 (119905 119909) = 0 119894 isinN 119905 ⩾ 0 119909 isin 120597Ω

(54b)


0 5 10 15 20 25 300

0005

001

0015

002


u1(t 0261)

0 5 10 15 20 25 300

02

04

06

08

10

0005

001

0015

002

0025

Time t


Spacex

u1

20 25 30

02

04

6

8



120601 (119904 119909) = (02 cos (2120587119909 (1 minus 119909))2 sin (120587119904)2

025 sin (6119909 (1 minus 119909))2 cos (3120587119904)2)

minus 83 le 119904 le 0

(55)

where 119909 isin Ω ≜ (1199091 1199092)119879isin 119877

2 |119909119894| lt 1 119894 isin N119873 = 1 2

119903 isin 119878 ≜ 1 2 119888111 = 119888121 = 01 119888211 = 119888

221 = 011 119888112 = 119888

122 =

015 119888212 = 012 = 119888222 119889

111 = 119889

121 = 01 119889211 = 119889

221 = 011

119889112 = 119889

122 = 015 119889212 = 012 = 119889

222 1198871(1199061) = 119906

31 + (12)1199061

1198872(1199062) = 11990632 + (13)1199062 and 1198611 = 12 and 1198612 = 13 Consider


119903

119894 119903 isin 119878 Let ℎ11 =

1 ℎ12 = 2 ℎ21 = 3 ℎ22 = 4 and 120573 = 001 Denote 119886119894(119903) = 119886119903

119894

and 119887119894(119903) = 119887119903



minus 1198861 +1198871

1 minus 119870119890120573120591= minus 09570 lt 0

minus 1198862 +1198872

1 minus 119870119890120573120591= minus 00084 lt 0

(56)



5 Conclusions





Acknowledgments



0 1 2 3 4 5 6 7 80

0002

0004

0006

0008

001

0012

0014

0016

0018


u2(t 0258)

0 1 2 3 4 5 6 7 8002

0406

081

0

0005

001

0015

002

0025

003

Time t


Spacex

u2

5 6 7002

046



References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










and hence

1003816100381610038161003816119891119894 (119904)1003816100381610038161003816 +1003816100381610038161003816119892119894 (119904)

1003816100381610038161003816 ⩽ 119887119872 + |119904|119902minus1

forall119904 isin 119877 (19)



1003816100381610038161003816F119894 (119904)1003816100381610038161003816 ⩽ int

119904

0

1003816100381610038161003816119891119894 (119910)1003816100381610038161003816 119889119910 ⩽ 119887119872119904 +

1119902119904119902= 119887119872 |119904| +

1119902|119904|

119902 (20)



1003816100381610038161003816F119894 (119904)1003816100381610038161003816 ⩽ int

0

119904

1003816100381610038161003816119891119894 (119910)1003816100381610038161003816 119889119910 ⩽ int

0

119904

(119887119872 +10038161003816100381610038161199101003816100381610038161003816119902minus1) 119889119910

⩽ 119887119872 |119904| +1119902|119904|

119902

(21)

Hence

1003816100381610038161003816F119894 (119904)1003816100381610038161003816 ⩽ 119887119872 |119904| +

1119902|119904|

119902 forall119904 isin 119877 (22)


1003816100381610038161003816G119894 (119904)1003816100381610038161003816 ⩽ 119887119872 |119904| +

1119902|119904|

119902 forall119904 isin 119877 (23)

Denote |119888119894119895| = max119903isin119878|119888119903

119894119895| and |119889119894119895| = max119903isin119878|119889

119903

119894119895| then


119869

sum

119895=1120588119895 (120596 (119905)) (119888

119903

119894119895F119894 (119904) + 119889

119903

119894119895G119894 (119904))

⩽ (

119869

sum

119895=1

1003816100381610038161003816100381611988811989411989510038161003816100381610038161003816)1003816100381610038161003816F119894 (119904)

1003816100381610038161003816 +(

119869

sum

119895=1

10038161003816100381610038161003816119889119894119895

10038161003816100381610038161003816)1003816100381610038161003816G119894 (119904)

1003816100381610038161003816

⩽ 1198861198941 |119904| + 1198861198942 |119904|119902 forall119904 isin 119877

(24)

where 1198861198941 = 119887119872(sum119869

119895=1 |119888119894119895| + sum119869

119895=1 |119889119894119895|) and 1198861198942 =

(1119902)(sum119869

119895=1 |119888119894119895| + sum119869


Thereby we have

[

[

119869

sum

119895=1120588119895 (120596 (119905)) (119888

119903

119894119895F119894 (119904) + 119889

119903

119894119895G119894 (119904))

]

]

+ 119868119894119904

⩽ (1198861198941 +119872) |119904| + 1198861198942 |119904|119902 forall119904 isin 119877

(25)





1199021198861198942


120593119894 (120578) ⩾1119901D119894

10038171003817100381710038171205781003817100381710038171003817119901minus (1198861198941 +119872)int

Ω

10038161003816100381610038161205781003816100381610038161003816 119889119909

minus 1198861198942 intΩ

10038161003816100381610038161205781003816100381610038161003816119902119889119909

⩾1119901D119894

10038171003817100381710038171205781003817100381710038171003817119901minus 1198871198942

10038171003817100381710038171205781003817100381710038171003817119902minus 1198871198941

10038171003817100381710038171205781003817100381710038171003817 119894 isinN

(26)

Denote120583(119904) = (1119901)D119894119904119901minus1198871198942119904




120583 (119904) 119904 isin 119877 (27)


11199010 (Ω)


lowast as follows

⟨A119894 (120578) 120585⟩ = D119894 intΩ

1003816100381610038161003816nabla1205781003816100381610038161003816119901minus2

nabla120578nabla120585 119889119909

⟨B119894 (120578) 120585⟩ = intΩ

(119887119894 (120578)

minus

119869

sum

119895=1120588119895 (120596 (119905)) (119888

119903

119894119895119891119894 (120578) + 119889

119903

119894119895119892119894 (120578)) minus 119868119894)120585119889119909

(28)


1003816100381610038161003816119887119894 (119904)1003816100381610038161003816 +1003816100381610038161003816119891119894 (119904)

1003816100381610038161003816 +1003816100381610038161003816119892119894 (119904)

1003816100381610038161003816 ⩽ 119888119872 + |119904|119902minus1

forall119904 isin 119877 (29)


10038161003816100381610038161003816100381610038161003816100381610038161003816

119887119894 (119904) minus

119869

sum

119895=1120588119895 (120596 (119905)) (119888

119903

119894119895119891119894 (119904) + 119889

119903

119894119895119892119894 (119904)) minus 119868119894

10038161003816100381610038161003816100381610038161003816100381610038161003816

⩽ 119888119872 + |119904|119902minus1

+

119869

sum

119895=1

1003816100381610038161003816100381611988811989411989510038161003816100381610038161003816sdot1003816100381610038161003816119891119894 (119904)

1003816100381610038161003816 +(

119869

sum

119895=1

10038161003816100381610038161003816119889119903

119894119895

10038161003816100381610038161003816)

sdot1003816100381610038161003816119892119894 (119904)

1003816100381610038161003816 +119872 ⩽ 1198861198943 + 1198861198944 |119904|119902minus1

119904 isin 119877

(30)

Here 1198861198943 = 119872 + 119888119872 + 119888119872(sum119869

119895=1 |119888119894119895| + sum119869

119895=1 |119889119894119895|) and 1198861198944 =

1 + sum119869

119895=1 |119888119894119895| + sum119869


119894are



is bounded



⟨1205931015840


Owing to 1205931015840








120593119894 (120578119899) ⩾1119901D119894

10038171003817100381710038171205781198991003817100381710038171003817119901minus 1198871198942

10038171003817100381710038171205781198991003817100381710038171003817119902minus 1198871198941

10038171003817100381710038171205781198991003817100381710038171003817 (32)



dist (120578 120585) = 1003817100381710038171003817120578 minus 1205851003817100381710038171003817 (33)







120593119894 (120578) gt 120593119894 (120578119899) minus1119899

1003817100381710038171003817120578 minus 1205781198991003817100381710038171003817 forall120578 = 120578119899 (34)

120593119894 (120578119899) lt 119888119894 +1119899 (35)


100381710038171003817100381710038171205931015840

119894(120578119899)

10038171003817100381710038171003817= sup

120585=1

1003816100381610038161003816119889120593119894 (120578119899 120585)1003816100381610038161003816 ⩽

1119899 (36)


119894(120578119899) is



lowast

119894isin 119882

11199010 (Ω)



120593119894(119906lowast


lowast

119894(119909) in119883 for all 119894 isin


lowast

1 119906lowast

2 119906lowast

119899)119879

for (4)



119881119894 (119905 119903 (119905)) = 1198811119894 (119905 119903 (119905)) +1198812119894 (119905 119903 (119905)) 119894 isinN (37)

where

1198811119894 (119905 119903 (119905)) = 119890120573119905intΩ

ℎ119894 (119903 (119905))1003816100381610038161003816119906119894 (119905 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119909

1198812119894 (119905 119903 (119905))

=119887119894

1 minus 119870119890120573120591intΩ

int

119905

119905minus120591119894(119905)

119890120573119904 1003816100381610038161003816119906119894 (119904 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119904 119889119909

(38)


L1198811119894 (119905 119903)

= lim120575rarr 0+

1120575E [int

Ω

119890120573(119905+120575)

ℎ119894 (119903 (119905 + 120575))1003816100381610038161003816119906119894 (119905 + 120575 119909) minus 119906

lowast

119894

1003816100381610038161003816119901119889119909 |

119903 (119905) = 119903] minus 119890120573119905intΩ

ℎ119894 (119903 (119905))1003816100381610038161003816119906119894 (119905 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119909

= 119890120573119905

intΩ

ℎ119894 (119903 (119905)) 1199011003816100381610038161003816119906119894 minus119906

lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894)

sdot [

[

D119894

119898

sum

119896=1(120597

120597119909119896

(1003816100381610038161003816nabla119906119894

1003816100381610038161003816119901minus2 120597119906119894

120597119909119896

)minus120597

120597119909119896

(1003816100381610038161003816nabla119906

lowast

119894

1003816100381610038161003816119901minus2 120597119906

lowast

119894

120597119909119896

))


119894)) +

119869

sum

119895=1120588119895 (120596 (119905)) 119888

119903

119894119895(119891119894 (119906119894) minus119891119894 (119906

lowast

119894))

+

119869

sum

119895=1120588119895 (120596 (119905)) 119889

119903

119894119895(119892119894 (119906119894 (119905 minus 120591119894 (119905) 119909)) minus 119892119894 (119906

lowast

119894))]

]

119889119909

+intΩ

sum

119896isin119878

120587119894119896ℎ119894 (119896)1003816100381610038161003816119906119894 (119905 119909) minus 119906

lowast

119894

1003816100381610038161003816119901119889119909+120573ℎ119894 (119903 (119905)) int

Ω

1003816100381610038161003816119906119894

minus119906lowast

119894

1003816100381610038161003816119901119889119909

(39)

Next we claim that

119898

sum

119896=1intΩ

D119894

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894) (

120597

120597119909119896

(1003816100381610038161003816nabla119906119894

1003816100381610038161003816119901minus2 120597119906119894

120597119909119896

)

minus120597

120597119909119896

(1003816100381610038161003816nabla119906

lowast

119894

1003816100381610038161003816119901minus2 120597119906

lowast

119894

120597119909119896

))119889119909 ⩽ 0

(40)



(1003816100381610038161003816nabla120578

1003816100381610038161003816119901minus2


1003816100381610038161003816119901minus2




1003816100381610038161003816nabla1205781003816100381610038161003816119901minus2


1003816100381610038161003816119901minus1 1003816100381610038161003816nabla120595

1003816100381610038161003816

⩽119901 minus 1119901

(1003816100381610038161003816nabla120578

1003816100381610038161003816119901minus1)119901(119901minus1)

+

1003816100381610038161003816nabla1205951003816100381610038161003816119901

119901

1003816100381610038161003816nabla1205951003816100381610038161003816119901minus2


(1003816100381610038161003816nabla120595

1003816100381610038161003816119901minus1)119901(119901minus1)

+

1003816100381610038161003816nabla1205781003816100381610038161003816119901

119901

(42)



119898

sum

119896=1intΩ

D119894

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894)

sdot (120597

120597119909119896

(1003816100381610038161003816nabla119906119894

1003816100381610038161003816119901minus2 120597119906119894

120597119909119896

)minus120597

120597119909119896

(1003816100381610038161003816nabla119906

lowast

119894

1003816100381610038161003816119901minus2 120597119906

lowast

119894

120597119909119896

))119889119909

= minus

119898

sum

119896=1intΩ

D119894 (1003816100381610038161003816nabla119906119894

1003816100381610038161003816119901minus2 120597119906119894

120597119909119896

minus1003816100381610038161003816nabla119906

lowast

119894

1003816100381610038161003816119901minus2 120597119906

lowast

119894

120597119909119896

)

sdot120597

120597119909119896

(1003816100381610038161003816119906119894 minus 119906

lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus 119906lowast

119894)) 119889119909 = minusint

Ω

D119894 (119901


119894

1003816100381610038161003816119901minus2

[(1003816100381610038161003816nabla119906119894

1003816100381610038161003816119901minus2


lowast

119894

1003816100381610038161003816119901minus2

nabla119906lowast

119894)


119894)] 119889119909 ⩽ 0

(43)

which proves (40)


intΩ

ℎ119894 (119903 (119905)) 1199011003816100381610038161003816119906119894 minus119906

lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894)


119894)) 119889119909 ⩾ ℎ119894 (119903 (119905))

sdot 119901119861119894 intΩ

1003816100381610038161003816119906i minus119906lowast

119894

1003816100381610038161003816119901119889119909

(44)


intΩ

ℎ119894 (119903 (119905)) 1199011003816100381610038161003816119906119894 minus119906

lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894)

sdot

119869

sum

119895=1120588119895 (120596 (119905)) 119888

119903

119894119895(119891119894 (119906119894) minus119891119894 (119906

lowast

119894)) 119889119909 ⩽ ℎ119894 (119903 (119905))

sdot 119901119865119894(

119869

sum

119895=1

10038161003816100381610038161003816119888119903

119894119895

10038161003816100381610038161003816)int

Ω

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901119889119909

(45)


intΩ

ℎ119894 (119903 (119905)) 1199011003816100381610038161003816119906119894 minus119906

lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894)

sdot

119869

sum

119895=1120588119895 (120596 (119905)) 119889

119903

119894119895(119892119894 (119906119894 (119905 minus 120591119894 (119905) 119909)) minus 119892119894 (119906

lowast

119894))

⩽ ℎ119894 (119903 (119905)) 119866119894(

119869

sum

119895=1

10038161003816100381610038161003816119889119903

119894119895

10038161003816100381610038161003816) [(119901 minus 1) int

Ω

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901119889119909

+intΩ


119894

1003816100381610038161003816119901119889119909]

(46)


L1198811119894 (119905 119903) ⩽ 119890120573119905[minus119886119894 (119903) int

Ω

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901119889119909

+ 119887119894 (119903) intΩ


119894

1003816100381610038161003816119901119889119909] forall119903 isin 119878

(47)


L1198812119894 ⩽119887119894

1 minus 119870119890120573120591intΩ

119890120573119905 1003816100381610038161003816119906119894 (119905 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119909

minus 119887119894 intΩ

119890120573119905 1003816100381610038161003816119906119894 (119905 minus 120591119894 (119905) 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119909

(48)

Thus we get by (14)

L119881119894 (119905 119903 (119905)) ⩽ 119890120573119905(minus119886119894 +

119887119894

1 minus 119870119890120573120591)int

Ω

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901119889119909

⩽ 0 forall119905 ⩾ 0(49)


119864119881 (119905 119903 (119905)) minus 119864119881 (0 119903 (0)) = 119864int119905

0L119881 (119904 119903 (119904)) 119889119904

⩽ 0 119905 ⩾ 0(50)

Hence we have

(min119903isin119878

ℎ119894 (119903 (119905))) 119864 (119890120573119905intΩ

1003816100381610038161003816119906119894 (119905 119909) minus 119906lowast

119894(119909)1003816100381610038161003816119901119889119909)

⩽ 119864119881 (119905 119903 (119905)) ⩽ 119864119881 (0 119903 (0))

⩽ 119864(max119903isin119878

ℎ119894 (119903 (119905)) intΩ

1003816100381610038161003816119906119894 (0 119909) minus 119906lowast

119894(119909)1003816100381610038161003816119901119889119909

+119887119894

1 minus 119870119890120573120591intΩ

int

0

minus120591

119890120573119904 1003816100381610038161003816119906119894 (119904 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119904 119889119909)

⩽ (max119903isin119878

ℎ119894 (119903 (119905)) +119887119894

1 minus 119870119890120573120591)


1198641003817100381710038171003817120601119894 (119904) minus 119906

lowast

119894

1003817100381710038171003817119901

119871119901

(51)


which implies

119864(intΩ

1003816100381610038161003816119906119894 (119905 119909) minus 119906lowast

119894(119909)1003816100381610038161003816119901119889119909)

⩽ ℎ119894119890minus120573119905 sup

minus120591⩽119904⩽01198641003817100381710038171003817120601119894 (119904) minus 119906

lowast

119894

1003817100381710038171003817119901

119871119901 119894 isinN

(52)



1198641003817100381710038171003817119906 minus 119906

lowast1003817100381710038171003817119901

119871119901= sum

119894isinN

119864(intΩ

1003816100381610038161003816119906119894 (119905 119909) minus 119906lowast

119894(119909)1003816100381610038161003816119901119889119909)

⩽ (max119894isinN

ℎ119894) 119890minus1205731199051198641003817100381710038171003817120601 minus 119906

lowast1003817100381710038171003817119901

120591

(53)



minus120591⩽119904⩽0(119864120601119894(119904) minus

119906lowast

119894119901

119871119901)



12([1199050minus120591 119879)times119877





= (119906lowast

1 119906lowast

2 119906lowast

119894 119906

lowast


119894


(119909) = (119906lowast

1 (119909) 119906lowast

2 (119909) 119906lowast

119894(119909) 119906

lowast

119899(119909)) for





4 Numerical Example



1198891199061 (119905 119909)

119889119905

= D1div (1003816100381610038161003816nabla1199061

10038161003816100381610038163nabla1199061) minus 1198871 (1199061)

+ (119888119903

111198911 (1199061 (119905 119909)) + 119889119903

111198921 (1199061 (119905 minus 1205911 (119905) 119909)))

+ 1198681 119905 gt 0 119909 isin Ω

1198891199062 (119905 119909)

119889119905

= D2div (1003816100381610038161003816nabla1199062

10038161003816100381610038163nabla1199062) minus 1198872 (1199062)

+ (119888119903

211198912 (1199062 (119905 119909)) + 119889119903

211198922 (1199062 (119905 minus 1205912 (119905) 119909)))

+ 1198682 119905 gt 0 119909 isin Ω

119906119894 (119905 119909) = 0 119894 isinN 119905 ⩾ 0 119909 isin 120597Ω

(54a)


1198891199061 (119905 119909)

119889119905

= D1div (1003816100381610038161003816nabla1199061

10038161003816100381610038163nabla1199061) minus 1198871 (1199061)

+ (119888119903

121198911 (1199061 (119905 119909)) + 119889119903

121198921 (1199061 (119905 minus 1205911 (119905) 119909)))

+ 1198681 119905 gt 0 119909 isin Ω

1198891199062 (119905 119909)

119889119905

= D2div (1003816100381610038161003816nabla1199062

10038161003816100381610038163nabla1199062) minus 1198872 (1199062)

+ (119888119903

221198912 (1199062 (119905 119909)) + 119889119903

221198922 (1199062 (119905 minus 1205912 (119905) 119909)))

+ 1198682 119905 gt 0 119909 isin Ω

119906119894 (119905 119909) = 0 119894 isinN 119905 ⩾ 0 119909 isin 120597Ω

(54b)


0 5 10 15 20 25 300

0005

001

0015

002


u1(t 0261)

0 5 10 15 20 25 300

02

04

06

08

10

0005

001

0015

002

0025

Time t


Spacex

u1

20 25 30

02

04

6

8



120601 (119904 119909) = (02 cos (2120587119909 (1 minus 119909))2 sin (120587119904)2

025 sin (6119909 (1 minus 119909))2 cos (3120587119904)2)

minus 83 le 119904 le 0

(55)

where 119909 isin Ω ≜ (1199091 1199092)119879isin 119877

2 |119909119894| lt 1 119894 isin N119873 = 1 2

119903 isin 119878 ≜ 1 2 119888111 = 119888121 = 01 119888211 = 119888

221 = 011 119888112 = 119888

122 =

015 119888212 = 012 = 119888222 119889

111 = 119889

121 = 01 119889211 = 119889

221 = 011

119889112 = 119889

122 = 015 119889212 = 012 = 119889

222 1198871(1199061) = 119906

31 + (12)1199061

1198872(1199062) = 11990632 + (13)1199062 and 1198611 = 12 and 1198612 = 13 Consider


119903

119894 119903 isin 119878 Let ℎ11 =

1 ℎ12 = 2 ℎ21 = 3 ℎ22 = 4 and 120573 = 001 Denote 119886119894(119903) = 119886119903

119894

and 119887119894(119903) = 119887119903



minus 1198861 +1198871

1 minus 119870119890120573120591= minus 09570 lt 0

minus 1198862 +1198872

1 minus 119870119890120573120591= minus 00084 lt 0

(56)



5 Conclusions





Acknowledgments



0 1 2 3 4 5 6 7 80

0002

0004

0006

0008

001

0012

0014

0016

0018


u2(t 0258)

0 1 2 3 4 5 6 7 8002

0406

081

0

0005

001

0015

002

0025

003

Time t


Spacex

u2

5 6 7002

046



References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











⟨1205931015840


Owing to 1205931015840








120593119894 (120578119899) ⩾1119901D119894

10038171003817100381710038171205781198991003817100381710038171003817119901minus 1198871198942

10038171003817100381710038171205781198991003817100381710038171003817119902minus 1198871198941

10038171003817100381710038171205781198991003817100381710038171003817 (32)



dist (120578 120585) = 1003817100381710038171003817120578 minus 1205851003817100381710038171003817 (33)







120593119894 (120578) gt 120593119894 (120578119899) minus1119899

1003817100381710038171003817120578 minus 1205781198991003817100381710038171003817 forall120578 = 120578119899 (34)

120593119894 (120578119899) lt 119888119894 +1119899 (35)


100381710038171003817100381710038171205931015840

119894(120578119899)

10038171003817100381710038171003817= sup

120585=1

1003816100381610038161003816119889120593119894 (120578119899 120585)1003816100381610038161003816 ⩽

1119899 (36)


119894(120578119899) is



lowast

119894isin 119882

11199010 (Ω)



120593119894(119906lowast


lowast

119894(119909) in119883 for all 119894 isin


lowast

1 119906lowast

2 119906lowast

119899)119879

for (4)



119881119894 (119905 119903 (119905)) = 1198811119894 (119905 119903 (119905)) +1198812119894 (119905 119903 (119905)) 119894 isinN (37)

where

1198811119894 (119905 119903 (119905)) = 119890120573119905intΩ

ℎ119894 (119903 (119905))1003816100381610038161003816119906119894 (119905 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119909

1198812119894 (119905 119903 (119905))

=119887119894

1 minus 119870119890120573120591intΩ

int

119905

119905minus120591119894(119905)

119890120573119904 1003816100381610038161003816119906119894 (119904 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119904 119889119909

(38)


L1198811119894 (119905 119903)

= lim120575rarr 0+

1120575E [int

Ω

119890120573(119905+120575)

ℎ119894 (119903 (119905 + 120575))1003816100381610038161003816119906119894 (119905 + 120575 119909) minus 119906

lowast

119894

1003816100381610038161003816119901119889119909 |

119903 (119905) = 119903] minus 119890120573119905intΩ

ℎ119894 (119903 (119905))1003816100381610038161003816119906119894 (119905 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119909

= 119890120573119905

intΩ

ℎ119894 (119903 (119905)) 1199011003816100381610038161003816119906119894 minus119906

lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894)

sdot [

[

D119894

119898

sum

119896=1(120597

120597119909119896

(1003816100381610038161003816nabla119906119894

1003816100381610038161003816119901minus2 120597119906119894

120597119909119896

)minus120597

120597119909119896

(1003816100381610038161003816nabla119906

lowast

119894

1003816100381610038161003816119901minus2 120597119906

lowast

119894

120597119909119896

))


119894)) +

119869

sum

119895=1120588119895 (120596 (119905)) 119888

119903

119894119895(119891119894 (119906119894) minus119891119894 (119906

lowast

119894))

+

119869

sum

119895=1120588119895 (120596 (119905)) 119889

119903

119894119895(119892119894 (119906119894 (119905 minus 120591119894 (119905) 119909)) minus 119892119894 (119906

lowast

119894))]

]

119889119909

+intΩ

sum

119896isin119878

120587119894119896ℎ119894 (119896)1003816100381610038161003816119906119894 (119905 119909) minus 119906

lowast

119894

1003816100381610038161003816119901119889119909+120573ℎ119894 (119903 (119905)) int

Ω

1003816100381610038161003816119906119894

minus119906lowast

119894

1003816100381610038161003816119901119889119909

(39)

Next we claim that

119898

sum

119896=1intΩ

D119894

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894) (

120597

120597119909119896

(1003816100381610038161003816nabla119906119894

1003816100381610038161003816119901minus2 120597119906119894

120597119909119896

)

minus120597

120597119909119896

(1003816100381610038161003816nabla119906

lowast

119894

1003816100381610038161003816119901minus2 120597119906

lowast

119894

120597119909119896

))119889119909 ⩽ 0

(40)



(1003816100381610038161003816nabla120578

1003816100381610038161003816119901minus2


1003816100381610038161003816119901minus2




1003816100381610038161003816nabla1205781003816100381610038161003816119901minus2


1003816100381610038161003816119901minus1 1003816100381610038161003816nabla120595

1003816100381610038161003816

⩽119901 minus 1119901

(1003816100381610038161003816nabla120578

1003816100381610038161003816119901minus1)119901(119901minus1)

+

1003816100381610038161003816nabla1205951003816100381610038161003816119901

119901

1003816100381610038161003816nabla1205951003816100381610038161003816119901minus2


(1003816100381610038161003816nabla120595

1003816100381610038161003816119901minus1)119901(119901minus1)

+

1003816100381610038161003816nabla1205781003816100381610038161003816119901

119901

(42)



119898

sum

119896=1intΩ

D119894

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894)

sdot (120597

120597119909119896

(1003816100381610038161003816nabla119906119894

1003816100381610038161003816119901minus2 120597119906119894

120597119909119896

)minus120597

120597119909119896

(1003816100381610038161003816nabla119906

lowast

119894

1003816100381610038161003816119901minus2 120597119906

lowast

119894

120597119909119896

))119889119909

= minus

119898

sum

119896=1intΩ

D119894 (1003816100381610038161003816nabla119906119894

1003816100381610038161003816119901minus2 120597119906119894

120597119909119896

minus1003816100381610038161003816nabla119906

lowast

119894

1003816100381610038161003816119901minus2 120597119906

lowast

119894

120597119909119896

)

sdot120597

120597119909119896

(1003816100381610038161003816119906119894 minus 119906

lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus 119906lowast

119894)) 119889119909 = minusint

Ω

D119894 (119901


119894

1003816100381610038161003816119901minus2

[(1003816100381610038161003816nabla119906119894

1003816100381610038161003816119901minus2


lowast

119894

1003816100381610038161003816119901minus2

nabla119906lowast

119894)


119894)] 119889119909 ⩽ 0

(43)

which proves (40)


intΩ

ℎ119894 (119903 (119905)) 1199011003816100381610038161003816119906119894 minus119906

lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894)


119894)) 119889119909 ⩾ ℎ119894 (119903 (119905))

sdot 119901119861119894 intΩ

1003816100381610038161003816119906i minus119906lowast

119894

1003816100381610038161003816119901119889119909

(44)


intΩ

ℎ119894 (119903 (119905)) 1199011003816100381610038161003816119906119894 minus119906

lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894)

sdot

119869

sum

119895=1120588119895 (120596 (119905)) 119888

119903

119894119895(119891119894 (119906119894) minus119891119894 (119906

lowast

119894)) 119889119909 ⩽ ℎ119894 (119903 (119905))

sdot 119901119865119894(

119869

sum

119895=1

10038161003816100381610038161003816119888119903

119894119895

10038161003816100381610038161003816)int

Ω

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901119889119909

(45)


intΩ

ℎ119894 (119903 (119905)) 1199011003816100381610038161003816119906119894 minus119906

lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894)

sdot

119869

sum

119895=1120588119895 (120596 (119905)) 119889

119903

119894119895(119892119894 (119906119894 (119905 minus 120591119894 (119905) 119909)) minus 119892119894 (119906

lowast

119894))

⩽ ℎ119894 (119903 (119905)) 119866119894(

119869

sum

119895=1

10038161003816100381610038161003816119889119903

119894119895

10038161003816100381610038161003816) [(119901 minus 1) int

Ω

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901119889119909

+intΩ


119894

1003816100381610038161003816119901119889119909]

(46)


L1198811119894 (119905 119903) ⩽ 119890120573119905[minus119886119894 (119903) int

Ω

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901119889119909

+ 119887119894 (119903) intΩ


119894

1003816100381610038161003816119901119889119909] forall119903 isin 119878

(47)


L1198812119894 ⩽119887119894

1 minus 119870119890120573120591intΩ

119890120573119905 1003816100381610038161003816119906119894 (119905 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119909

minus 119887119894 intΩ

119890120573119905 1003816100381610038161003816119906119894 (119905 minus 120591119894 (119905) 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119909

(48)

Thus we get by (14)

L119881119894 (119905 119903 (119905)) ⩽ 119890120573119905(minus119886119894 +

119887119894

1 minus 119870119890120573120591)int

Ω

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901119889119909

⩽ 0 forall119905 ⩾ 0(49)


119864119881 (119905 119903 (119905)) minus 119864119881 (0 119903 (0)) = 119864int119905

0L119881 (119904 119903 (119904)) 119889119904

⩽ 0 119905 ⩾ 0(50)

Hence we have

(min119903isin119878

ℎ119894 (119903 (119905))) 119864 (119890120573119905intΩ

1003816100381610038161003816119906119894 (119905 119909) minus 119906lowast

119894(119909)1003816100381610038161003816119901119889119909)

⩽ 119864119881 (119905 119903 (119905)) ⩽ 119864119881 (0 119903 (0))

⩽ 119864(max119903isin119878

ℎ119894 (119903 (119905)) intΩ

1003816100381610038161003816119906119894 (0 119909) minus 119906lowast

119894(119909)1003816100381610038161003816119901119889119909

+119887119894

1 minus 119870119890120573120591intΩ

int

0

minus120591

119890120573119904 1003816100381610038161003816119906119894 (119904 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119904 119889119909)

⩽ (max119903isin119878

ℎ119894 (119903 (119905)) +119887119894

1 minus 119870119890120573120591)


1198641003817100381710038171003817120601119894 (119904) minus 119906

lowast

119894

1003817100381710038171003817119901

119871119901

(51)


which implies

119864(intΩ

1003816100381610038161003816119906119894 (119905 119909) minus 119906lowast

119894(119909)1003816100381610038161003816119901119889119909)

⩽ ℎ119894119890minus120573119905 sup

minus120591⩽119904⩽01198641003817100381710038171003817120601119894 (119904) minus 119906

lowast

119894

1003817100381710038171003817119901

119871119901 119894 isinN

(52)



1198641003817100381710038171003817119906 minus 119906

lowast1003817100381710038171003817119901

119871119901= sum

119894isinN

119864(intΩ

1003816100381610038161003816119906119894 (119905 119909) minus 119906lowast

119894(119909)1003816100381610038161003816119901119889119909)

⩽ (max119894isinN

ℎ119894) 119890minus1205731199051198641003817100381710038171003817120601 minus 119906

lowast1003817100381710038171003817119901

120591

(53)



minus120591⩽119904⩽0(119864120601119894(119904) minus

119906lowast

119894119901

119871119901)



12([1199050minus120591 119879)times119877





= (119906lowast

1 119906lowast

2 119906lowast

119894 119906

lowast


119894


(119909) = (119906lowast

1 (119909) 119906lowast

2 (119909) 119906lowast

119894(119909) 119906

lowast

119899(119909)) for





4 Numerical Example



1198891199061 (119905 119909)

119889119905

= D1div (1003816100381610038161003816nabla1199061

10038161003816100381610038163nabla1199061) minus 1198871 (1199061)

+ (119888119903

111198911 (1199061 (119905 119909)) + 119889119903

111198921 (1199061 (119905 minus 1205911 (119905) 119909)))

+ 1198681 119905 gt 0 119909 isin Ω

1198891199062 (119905 119909)

119889119905

= D2div (1003816100381610038161003816nabla1199062

10038161003816100381610038163nabla1199062) minus 1198872 (1199062)

+ (119888119903

211198912 (1199062 (119905 119909)) + 119889119903

211198922 (1199062 (119905 minus 1205912 (119905) 119909)))

+ 1198682 119905 gt 0 119909 isin Ω

119906119894 (119905 119909) = 0 119894 isinN 119905 ⩾ 0 119909 isin 120597Ω

(54a)


1198891199061 (119905 119909)

119889119905

= D1div (1003816100381610038161003816nabla1199061

10038161003816100381610038163nabla1199061) minus 1198871 (1199061)

+ (119888119903

121198911 (1199061 (119905 119909)) + 119889119903

121198921 (1199061 (119905 minus 1205911 (119905) 119909)))

+ 1198681 119905 gt 0 119909 isin Ω

1198891199062 (119905 119909)

119889119905

= D2div (1003816100381610038161003816nabla1199062

10038161003816100381610038163nabla1199062) minus 1198872 (1199062)

+ (119888119903

221198912 (1199062 (119905 119909)) + 119889119903

221198922 (1199062 (119905 minus 1205912 (119905) 119909)))

+ 1198682 119905 gt 0 119909 isin Ω

119906119894 (119905 119909) = 0 119894 isinN 119905 ⩾ 0 119909 isin 120597Ω

(54b)


0 5 10 15 20 25 300

0005

001

0015

002


u1(t 0261)

0 5 10 15 20 25 300

02

04

06

08

10

0005

001

0015

002

0025

Time t


Spacex

u1

20 25 30

02

04

6

8



120601 (119904 119909) = (02 cos (2120587119909 (1 minus 119909))2 sin (120587119904)2

025 sin (6119909 (1 minus 119909))2 cos (3120587119904)2)

minus 83 le 119904 le 0

(55)

where 119909 isin Ω ≜ (1199091 1199092)119879isin 119877

2 |119909119894| lt 1 119894 isin N119873 = 1 2

119903 isin 119878 ≜ 1 2 119888111 = 119888121 = 01 119888211 = 119888

221 = 011 119888112 = 119888

122 =

015 119888212 = 012 = 119888222 119889

111 = 119889

121 = 01 119889211 = 119889

221 = 011

119889112 = 119889

122 = 015 119889212 = 012 = 119889

222 1198871(1199061) = 119906

31 + (12)1199061

1198872(1199062) = 11990632 + (13)1199062 and 1198611 = 12 and 1198612 = 13 Consider


119903

119894 119903 isin 119878 Let ℎ11 =

1 ℎ12 = 2 ℎ21 = 3 ℎ22 = 4 and 120573 = 001 Denote 119886119894(119903) = 119886119903

119894

and 119887119894(119903) = 119887119903



minus 1198861 +1198871

1 minus 119870119890120573120591= minus 09570 lt 0

minus 1198862 +1198872

1 minus 119870119890120573120591= minus 00084 lt 0

(56)



5 Conclusions





Acknowledgments



0 1 2 3 4 5 6 7 80

0002

0004

0006

0008

001

0012

0014

0016

0018


u2(t 0258)

0 1 2 3 4 5 6 7 8002

0406

081

0

0005

001

0015

002

0025

003

Time t


Spacex

u2

5 6 7002

046



References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1003816100381610038161003816nabla1205781003816100381610038161003816119901minus2


1003816100381610038161003816119901minus1 1003816100381610038161003816nabla120595

1003816100381610038161003816

⩽119901 minus 1119901

(1003816100381610038161003816nabla120578

1003816100381610038161003816119901minus1)119901(119901minus1)

+

1003816100381610038161003816nabla1205951003816100381610038161003816119901

119901

1003816100381610038161003816nabla1205951003816100381610038161003816119901minus2


(1003816100381610038161003816nabla120595

1003816100381610038161003816119901minus1)119901(119901minus1)

+

1003816100381610038161003816nabla1205781003816100381610038161003816119901

119901

(42)



119898

sum

119896=1intΩ

D119894

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894)

sdot (120597

120597119909119896

(1003816100381610038161003816nabla119906119894

1003816100381610038161003816119901minus2 120597119906119894

120597119909119896

)minus120597

120597119909119896

(1003816100381610038161003816nabla119906

lowast

119894

1003816100381610038161003816119901minus2 120597119906

lowast

119894

120597119909119896

))119889119909

= minus

119898

sum

119896=1intΩ

D119894 (1003816100381610038161003816nabla119906119894

1003816100381610038161003816119901minus2 120597119906119894

120597119909119896

minus1003816100381610038161003816nabla119906

lowast

119894

1003816100381610038161003816119901minus2 120597119906

lowast

119894

120597119909119896

)

sdot120597

120597119909119896

(1003816100381610038161003816119906119894 minus 119906

lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus 119906lowast

119894)) 119889119909 = minusint

Ω

D119894 (119901


119894

1003816100381610038161003816119901minus2

[(1003816100381610038161003816nabla119906119894

1003816100381610038161003816119901minus2


lowast

119894

1003816100381610038161003816119901minus2

nabla119906lowast

119894)


119894)] 119889119909 ⩽ 0

(43)

which proves (40)


intΩ

ℎ119894 (119903 (119905)) 1199011003816100381610038161003816119906119894 minus119906

lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894)


119894)) 119889119909 ⩾ ℎ119894 (119903 (119905))

sdot 119901119861119894 intΩ

1003816100381610038161003816119906i minus119906lowast

119894

1003816100381610038161003816119901119889119909

(44)


intΩ

ℎ119894 (119903 (119905)) 1199011003816100381610038161003816119906119894 minus119906

lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894)

sdot

119869

sum

119895=1120588119895 (120596 (119905)) 119888

119903

119894119895(119891119894 (119906119894) minus119891119894 (119906

lowast

119894)) 119889119909 ⩽ ℎ119894 (119903 (119905))

sdot 119901119865119894(

119869

sum

119895=1

10038161003816100381610038161003816119888119903

119894119895

10038161003816100381610038161003816)int

Ω

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901119889119909

(45)


intΩ

ℎ119894 (119903 (119905)) 1199011003816100381610038161003816119906119894 minus119906

lowast

119894

1003816100381610038161003816119901minus2

(119906119894 minus119906lowast

119894)

sdot

119869

sum

119895=1120588119895 (120596 (119905)) 119889

119903

119894119895(119892119894 (119906119894 (119905 minus 120591119894 (119905) 119909)) minus 119892119894 (119906

lowast

119894))

⩽ ℎ119894 (119903 (119905)) 119866119894(

119869

sum

119895=1

10038161003816100381610038161003816119889119903

119894119895

10038161003816100381610038161003816) [(119901 minus 1) int

Ω

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901119889119909

+intΩ


119894

1003816100381610038161003816119901119889119909]

(46)


L1198811119894 (119905 119903) ⩽ 119890120573119905[minus119886119894 (119903) int

Ω

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901119889119909

+ 119887119894 (119903) intΩ


119894

1003816100381610038161003816119901119889119909] forall119903 isin 119878

(47)


L1198812119894 ⩽119887119894

1 minus 119870119890120573120591intΩ

119890120573119905 1003816100381610038161003816119906119894 (119905 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119909

minus 119887119894 intΩ

119890120573119905 1003816100381610038161003816119906119894 (119905 minus 120591119894 (119905) 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119909

(48)

Thus we get by (14)

L119881119894 (119905 119903 (119905)) ⩽ 119890120573119905(minus119886119894 +

119887119894

1 minus 119870119890120573120591)int

Ω

1003816100381610038161003816119906119894 minus119906lowast

119894

1003816100381610038161003816119901119889119909

⩽ 0 forall119905 ⩾ 0(49)


119864119881 (119905 119903 (119905)) minus 119864119881 (0 119903 (0)) = 119864int119905

0L119881 (119904 119903 (119904)) 119889119904

⩽ 0 119905 ⩾ 0(50)

Hence we have

(min119903isin119878

ℎ119894 (119903 (119905))) 119864 (119890120573119905intΩ

1003816100381610038161003816119906119894 (119905 119909) minus 119906lowast

119894(119909)1003816100381610038161003816119901119889119909)

⩽ 119864119881 (119905 119903 (119905)) ⩽ 119864119881 (0 119903 (0))

⩽ 119864(max119903isin119878

ℎ119894 (119903 (119905)) intΩ

1003816100381610038161003816119906119894 (0 119909) minus 119906lowast

119894(119909)1003816100381610038161003816119901119889119909

+119887119894

1 minus 119870119890120573120591intΩ

int

0

minus120591

119890120573119904 1003816100381610038161003816119906119894 (119904 119909) minus 119906

lowast

119894(119909)1003816100381610038161003816119901119889119904 119889119909)

⩽ (max119903isin119878

ℎ119894 (119903 (119905)) +119887119894

1 minus 119870119890120573120591)


1198641003817100381710038171003817120601119894 (119904) minus 119906

lowast

119894

1003817100381710038171003817119901

119871119901

(51)


which implies

119864(intΩ

1003816100381610038161003816119906119894 (119905 119909) minus 119906lowast

119894(119909)1003816100381610038161003816119901119889119909)

⩽ ℎ119894119890minus120573119905 sup

minus120591⩽119904⩽01198641003817100381710038171003817120601119894 (119904) minus 119906

lowast

119894

1003817100381710038171003817119901

119871119901 119894 isinN

(52)



1198641003817100381710038171003817119906 minus 119906

lowast1003817100381710038171003817119901

119871119901= sum

119894isinN

119864(intΩ

1003816100381610038161003816119906119894 (119905 119909) minus 119906lowast

119894(119909)1003816100381610038161003816119901119889119909)

⩽ (max119894isinN

ℎ119894) 119890minus1205731199051198641003817100381710038171003817120601 minus 119906

lowast1003817100381710038171003817119901

120591

(53)



minus120591⩽119904⩽0(119864120601119894(119904) minus

119906lowast

119894119901

119871119901)



12([1199050minus120591 119879)times119877





= (119906lowast

1 119906lowast

2 119906lowast

119894 119906

lowast


119894


(119909) = (119906lowast

1 (119909) 119906lowast

2 (119909) 119906lowast

119894(119909) 119906

lowast

119899(119909)) for





4 Numerical Example



1198891199061 (119905 119909)

119889119905

= D1div (1003816100381610038161003816nabla1199061

10038161003816100381610038163nabla1199061) minus 1198871 (1199061)

+ (119888119903

111198911 (1199061 (119905 119909)) + 119889119903

111198921 (1199061 (119905 minus 1205911 (119905) 119909)))

+ 1198681 119905 gt 0 119909 isin Ω

1198891199062 (119905 119909)

119889119905

= D2div (1003816100381610038161003816nabla1199062

10038161003816100381610038163nabla1199062) minus 1198872 (1199062)

+ (119888119903

211198912 (1199062 (119905 119909)) + 119889119903

211198922 (1199062 (119905 minus 1205912 (119905) 119909)))

+ 1198682 119905 gt 0 119909 isin Ω

119906119894 (119905 119909) = 0 119894 isinN 119905 ⩾ 0 119909 isin 120597Ω

(54a)


1198891199061 (119905 119909)

119889119905

= D1div (1003816100381610038161003816nabla1199061

10038161003816100381610038163nabla1199061) minus 1198871 (1199061)

+ (119888119903

121198911 (1199061 (119905 119909)) + 119889119903

121198921 (1199061 (119905 minus 1205911 (119905) 119909)))

+ 1198681 119905 gt 0 119909 isin Ω

1198891199062 (119905 119909)

119889119905

= D2div (1003816100381610038161003816nabla1199062

10038161003816100381610038163nabla1199062) minus 1198872 (1199062)

+ (119888119903

221198912 (1199062 (119905 119909)) + 119889119903

221198922 (1199062 (119905 minus 1205912 (119905) 119909)))

+ 1198682 119905 gt 0 119909 isin Ω

119906119894 (119905 119909) = 0 119894 isinN 119905 ⩾ 0 119909 isin 120597Ω

(54b)


0 5 10 15 20 25 300

0005

001

0015

002


u1(t 0261)

0 5 10 15 20 25 300

02

04

06

08

10

0005

001

0015

002

0025

Time t


Spacex

u1

20 25 30

02

04

6

8



120601 (119904 119909) = (02 cos (2120587119909 (1 minus 119909))2 sin (120587119904)2

025 sin (6119909 (1 minus 119909))2 cos (3120587119904)2)

minus 83 le 119904 le 0

(55)

where 119909 isin Ω ≜ (1199091 1199092)119879isin 119877

2 |119909119894| lt 1 119894 isin N119873 = 1 2

119903 isin 119878 ≜ 1 2 119888111 = 119888121 = 01 119888211 = 119888

221 = 011 119888112 = 119888

122 =

015 119888212 = 012 = 119888222 119889

111 = 119889

121 = 01 119889211 = 119889

221 = 011

119889112 = 119889

122 = 015 119889212 = 012 = 119889

222 1198871(1199061) = 119906

31 + (12)1199061

1198872(1199062) = 11990632 + (13)1199062 and 1198611 = 12 and 1198612 = 13 Consider


119903

119894 119903 isin 119878 Let ℎ11 =

1 ℎ12 = 2 ℎ21 = 3 ℎ22 = 4 and 120573 = 001 Denote 119886119894(119903) = 119886119903

119894

and 119887119894(119903) = 119887119903



minus 1198861 +1198871

1 minus 119870119890120573120591= minus 09570 lt 0

minus 1198862 +1198872

1 minus 119870119890120573120591= minus 00084 lt 0

(56)



5 Conclusions





Acknowledgments



0 1 2 3 4 5 6 7 80

0002

0004

0006

0008

001

0012

0014

0016

0018


u2(t 0258)

0 1 2 3 4 5 6 7 8002

0406

081

0

0005

001

0015

002

0025

003

Time t


Spacex

u2

5 6 7002

046



References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










which implies

119864(intΩ

1003816100381610038161003816119906119894 (119905 119909) minus 119906lowast

119894(119909)1003816100381610038161003816119901119889119909)

⩽ ℎ119894119890minus120573119905 sup

minus120591⩽119904⩽01198641003817100381710038171003817120601119894 (119904) minus 119906

lowast

119894

1003817100381710038171003817119901

119871119901 119894 isinN

(52)



1198641003817100381710038171003817119906 minus 119906

lowast1003817100381710038171003817119901

119871119901= sum

119894isinN

119864(intΩ

1003816100381610038161003816119906119894 (119905 119909) minus 119906lowast

119894(119909)1003816100381610038161003816119901119889119909)

⩽ (max119894isinN

ℎ119894) 119890minus1205731199051198641003817100381710038171003817120601 minus 119906

lowast1003817100381710038171003817119901

120591

(53)



minus120591⩽119904⩽0(119864120601119894(119904) minus

119906lowast

119894119901

119871119901)



12([1199050minus120591 119879)times119877





= (119906lowast

1 119906lowast

2 119906lowast

119894 119906

lowast


119894


(119909) = (119906lowast

1 (119909) 119906lowast

2 (119909) 119906lowast

119894(119909) 119906

lowast

119899(119909)) for





4 Numerical Example



1198891199061 (119905 119909)

119889119905

= D1div (1003816100381610038161003816nabla1199061

10038161003816100381610038163nabla1199061) minus 1198871 (1199061)

+ (119888119903

111198911 (1199061 (119905 119909)) + 119889119903

111198921 (1199061 (119905 minus 1205911 (119905) 119909)))

+ 1198681 119905 gt 0 119909 isin Ω

1198891199062 (119905 119909)

119889119905

= D2div (1003816100381610038161003816nabla1199062

10038161003816100381610038163nabla1199062) minus 1198872 (1199062)

+ (119888119903

211198912 (1199062 (119905 119909)) + 119889119903

211198922 (1199062 (119905 minus 1205912 (119905) 119909)))

+ 1198682 119905 gt 0 119909 isin Ω

119906119894 (119905 119909) = 0 119894 isinN 119905 ⩾ 0 119909 isin 120597Ω

(54a)


1198891199061 (119905 119909)

119889119905

= D1div (1003816100381610038161003816nabla1199061

10038161003816100381610038163nabla1199061) minus 1198871 (1199061)

+ (119888119903

121198911 (1199061 (119905 119909)) + 119889119903

121198921 (1199061 (119905 minus 1205911 (119905) 119909)))

+ 1198681 119905 gt 0 119909 isin Ω

1198891199062 (119905 119909)

119889119905

= D2div (1003816100381610038161003816nabla1199062

10038161003816100381610038163nabla1199062) minus 1198872 (1199062)

+ (119888119903

221198912 (1199062 (119905 119909)) + 119889119903

221198922 (1199062 (119905 minus 1205912 (119905) 119909)))

+ 1198682 119905 gt 0 119909 isin Ω

119906119894 (119905 119909) = 0 119894 isinN 119905 ⩾ 0 119909 isin 120597Ω

(54b)


0 5 10 15 20 25 300

0005

001

0015

002


u1(t 0261)

0 5 10 15 20 25 300

02

04

06

08

10

0005

001

0015

002

0025

Time t


Spacex

u1

20 25 30

02

04

6

8



120601 (119904 119909) = (02 cos (2120587119909 (1 minus 119909))2 sin (120587119904)2

025 sin (6119909 (1 minus 119909))2 cos (3120587119904)2)

minus 83 le 119904 le 0

(55)

where 119909 isin Ω ≜ (1199091 1199092)119879isin 119877

2 |119909119894| lt 1 119894 isin N119873 = 1 2

119903 isin 119878 ≜ 1 2 119888111 = 119888121 = 01 119888211 = 119888

221 = 011 119888112 = 119888

122 =

015 119888212 = 012 = 119888222 119889

111 = 119889

121 = 01 119889211 = 119889

221 = 011

119889112 = 119889

122 = 015 119889212 = 012 = 119889

222 1198871(1199061) = 119906

31 + (12)1199061

1198872(1199062) = 11990632 + (13)1199062 and 1198611 = 12 and 1198612 = 13 Consider


119903

119894 119903 isin 119878 Let ℎ11 =

1 ℎ12 = 2 ℎ21 = 3 ℎ22 = 4 and 120573 = 001 Denote 119886119894(119903) = 119886119903

119894

and 119887119894(119903) = 119887119903



minus 1198861 +1198871

1 minus 119870119890120573120591= minus 09570 lt 0

minus 1198862 +1198872

1 minus 119870119890120573120591= minus 00084 lt 0

(56)



5 Conclusions





Acknowledgments



0 1 2 3 4 5 6 7 80

0002

0004

0006

0008

001

0012

0014

0016

0018


u2(t 0258)

0 1 2 3 4 5 6 7 8002

0406

081

0

0005

001

0015

002

0025

003

Time t


Spacex

u2

5 6 7002

046



References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 20 25 300

0005

001

0015

002


u1(t 0261)

0 5 10 15 20 25 300

02

04

06

08

10

0005

001

0015

002

0025

Time t


Spacex

u1

20 25 30

02

04

6

8



120601 (119904 119909) = (02 cos (2120587119909 (1 minus 119909))2 sin (120587119904)2

025 sin (6119909 (1 minus 119909))2 cos (3120587119904)2)

minus 83 le 119904 le 0

(55)

where 119909 isin Ω ≜ (1199091 1199092)119879isin 119877

2 |119909119894| lt 1 119894 isin N119873 = 1 2

119903 isin 119878 ≜ 1 2 119888111 = 119888121 = 01 119888211 = 119888

221 = 011 119888112 = 119888

122 =

015 119888212 = 012 = 119888222 119889

111 = 119889

121 = 01 119889211 = 119889

221 = 011

119889112 = 119889

122 = 015 119889212 = 012 = 119889

222 1198871(1199061) = 119906

31 + (12)1199061

1198872(1199062) = 11990632 + (13)1199062 and 1198611 = 12 and 1198612 = 13 Consider


119903

119894 119903 isin 119878 Let ℎ11 =

1 ℎ12 = 2 ℎ21 = 3 ℎ22 = 4 and 120573 = 001 Denote 119886119894(119903) = 119886119903

119894

and 119887119894(119903) = 119887119903



minus 1198861 +1198871

1 minus 119870119890120573120591= minus 09570 lt 0

minus 1198862 +1198872

1 minus 119870119890120573120591= minus 00084 lt 0

(56)



5 Conclusions





Acknowledgments



0 1 2 3 4 5 6 7 80

0002

0004

0006

0008

001

0012

0014

0016

0018


u2(t 0258)

0 1 2 3 4 5 6 7 8002

0406

081

0

0005

001

0015

002

0025

003

Time t


Spacex

u2

5 6 7002

046



References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 6 7 80

0002

0004

0006

0008

001

0012

0014

0016

0018


u2(t 0258)

0 1 2 3 4 5 6 7 8002

0406

081

0

0005

001

0015

002

0025

003

Time t


Spacex

u2

5 6 7002

046



References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article existence of exponential -stability nonconstant...

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