nonlinear pdes in sobolev spaces with variable...

Preprint CMAF Pre-015 (2013)

http://cmaf.ptmat.fc.ul.pt/preprints.html

Nonlinear PDEs in Sobolev Spaces with Variable Exponents

Stanislav Antontsev and Sergey Shmarev

The slide presentation for the invited lecture given at the International Conference “DifferentialEquations, Function Spaces, Approximation Theory" dedicated to the 105th Anniversary of SergeySobolev, August 18-24, Novosibirsk, Russia

S. Antontsev, S. Shmarev PDEs with Variable Nonlinearity 1

Nonlinear PDEs in Sobolev spaces with variableexponents

Stanislav Antontsev 1 Sergey Shmarev 2

1CMAF, University of Lisbon

2University of Oviedo

Differential Equations, Function Spaces, Approximation TheoryAugust 18-24, Novosibirsk, Russia

International Conference Dedicated to 105th Anniversary of Sergey Sobolev


Physical motivation

Electro-rheological fluidsrot E = 0, div E = 0, div v = 0

−div a(x,E,D(v)) +∇P = div (v⊗ v) + f

a(x,E,D(v)) ≈ (1+ |D(v)|)(p(E)−2)/2D(v)

Non-Newtonian fluids (v · ∇)v = div(µ(θ) + τ(θ)|D(v)|p(θ)−2D(v)

)−∇P+ f)

div v = 0, −∆θ+ v · ∇b(θ) = g(x)

Filtration of an ideal barotropic gas in a non-homogeneous porous medium:

ρt +∑i

Di

(ρ |Dip|

λi(θ)−2Dip

)= h p = ργ(θ)

Processing of digital images, e.g., Perona-Malik anisotropic diffusion models,

ut = div (G(|∇u|)∇u)) G(s) = e−s ≈

1 as s→ 0

0 as s→∞enhances edges big |∇u| ⇒ slow diffusion


Lavrentiev’s phenomenon

The classical minimization problem J(u) =

∫Ωf(x,∇u)dx, u ∈W1,p0 (Ω), Ω ⊂ Rn

−C1 + C2|r|p 6 f(x, r) 6 C3 + C4|r|

p, p = const

The classical Sobolev spaces

minJ(u)| u ∈W1,p0 (Ω)

= inf

J(u)| u ∈ C∞

0 (Ω)

Minimization of a functional with nonstandard growth

−C1 + C2|r|p(x) 6 f(x, r) 6 C3 + C4|r|

p(x) 1 < p− 6 p(x) 6 p+ <∞Generalized Sobolev spaces: W1,p(x)0 (Ω)

A model problem:

J(u) =

∫Ω

[1

p(x)|∇u|p(x) + ug(x)

]dx u ∈W1,p(x)0 (Ω)

div(|∇u|p(x)−2∇u

)= g(x)

Lavrentiev’s phenomen: M.A. Lavrentiev “Sur quelques problemes du calcul devariations". Annali di Matematica Pura ed Applicata, v.4, (1926) 107-124

Variable exponents Sobolevspaces

PDEs with variablenonlinearity

New mathematical modelsin continuum mechanics


Main issues

ut =

n∑i=1

Di(|Diu|pi(z)−2Diu) + f(z,u) z = (x, t) ∈ Q = Ω× (0, T)

u(x, 0) = u0(x) in Ωu = 0 on Γ = ∂Ω× (0, T)

existence of weak (energy) solutions

space localization:

u0 ≡ 0 in ω ⊂ Ω ⇒ u(z) ≡ 0 in ω ′ × (0, τ), ω ′ ⊆ ω

vanishing in a finite time:

∃t∗ : ‖u‖2,Ω(t) = 0 ∀ t > t∗

blow-up:

∃t∗, ω ⊆ Ω : ‖u‖∞,ω(t)→∞ as t→ t∗−

the influence of anisotropy on the propagation properties of solutions.


Parabolic spaces

ut =

n∑i=1

Di(|Diu|pi(z)−2Diu) + f(z,u)

the main space

W =

u(x, t)∣∣∣∣∣∣∣u : [0, T ] 7→ Vt(Ω),

|u|2, |Diu|pi(x, t) ∈ L1(Q)

‖u‖W = ‖u‖2,Q +

n∑i=1

‖Diu‖pi(·,·),Q

the dual space W ′:

w ∈W′ ⇐⇒

w = w0,w1, . . . ,wn, w0 ∈ L2(Q), wi ∈ Lp′i(·)(Q),

∀φ ∈W 〈〈w,φ〉〉 =∫Q

(w0φ+

∑i

wiDiφ

)dz.


Basic properties

1 if p(x) ∈ C0(Ω), then

C∞(Ω) is dense in Lp(·)(Ω) W1,p(·)0 (Ω) is separable and reflexive

2 C∞0 (Ω) is dense in V(Ω) if p(x) is Log-continuous: ∀ z, ζ ∈ Ω, |z− ζ| < 1

∑i

|pi(z) − pi(ζ)| 6 ω(|z− ζ|), limτ→0+ ω(τ) ln

1

τ= C < +∞.

3 min(‖f‖p

−

p(·) , ‖f‖p+

p(·)

)6∫Ω

|f(x)|p(x) dx 6 max(‖f‖p

−

p(·) ,‖f‖p+

p(·)

)4 Hölder’s inequality: f ∈ Lp(·)(Ω), g ∈ Lp′(·)(Ω), p(x) ∈ (1,∞), p ′(x) =

p(x)

p(x) − 1∫Ω

|f g|dx 6

(1

p−+

1

(p′−)

)‖f‖p(·) ‖g‖p′(·) 6 2‖f‖p(·) ‖g‖p′(·)

L. Diening, P. Harjulehto, P. Hasto, and M. Ruzicka, Lebesgue and Sobolev Spaceswith Variable Exponents, Springer, Berlin, 2011. Lecture Notes in Mathematics, Vol. 2017,1st Edition.


Existence results for evolution p(x, t)-Laplace equation

Existence theorems for the evolution p(x, t) -laplacianut = div

(a(z,u)|∇u|p(x,t)−2∇u

)in Q

u = 0 on Γ , u(x, 0) = u0 in Ω0 < a− 6 a(z, s) 6 a+ <∞ (1)

1 S.Antontsev, S.Shm. (2006, 2009) - Galerkin’s approximations. Energy solution forp(x, t) continuous in Q with logarithmic module of continuity

2 Yu.Alkhutov, V.Zhikov (2010) - very weak solution for measurable and boundedp(x, t). The theory of monotone operators.

3 L.Diening, P. Nägele, M. Ružička (2011) Existence of energy solution for systems ofequations of the type (1). The theory of monotone operators + p(x, t) ∈ Clog(Q).

(For p independent of t: entropy and renormalized solutions with L1 data, variationalinequalities, equations with nonlocal terms, semi-group theory, stationary solutions,calculus of variations, .... )


Definition of energy solution

ut =

n∑i=1

Di(|Diu|pi(z)−2Diu) + f(z,u) in Q

u = 0 on Γ , u(x, 0) = u0 in Ω

(2)

Definitionu(x, t) is called energy solution of problem (2) if

1 u ∈W, ∂tu ∈W ′

2 ∀φ ∈W with ∂tφ ∈W ′

∫Q

(φ∂tu+

n∑i=1

|Diu|pi−2Diu ·Diφ− fφ

)dz = 0

3 ∀φ(x) ∈ C∞0 (Ω)

∫Ω(u(x, t) − u0(x))φ(x)dx→ 0 as t→ 0


Existence of energy solutions ut =

n∑i=1

Di(|Diu|pi(z)−2Diu) + f(z,u) in Q

u = 0 on Γ , u(x, 0) = u0 in Ω

(3)

Theorem (Global in time existence)

Let f(z, r) be a Carathéodory function, |f(z, r)| 6 c0|r|λ−1 + h(z), h ∈ Lλ′(Q) . Assume

that pi(z) are Log-continuous in Q. If one of the conditions

λ = max2,p− − δ with δ > 0, p− = mini

infQpi(z),

λ = max2,p− and c0 << 1

is fulfilled, then for every u0 ∈ L2(Ω) problem (3) has a solution u ∈W, ut ∈W ′.

Theorem (Local in time existence)Let |f(z,u)| 6 d0|u|λ−1 + h(z), λ > 2, h ∈ L1(0, T ;L∞(Ω)). Then ∀ u0 ∈ L∞(Ω) problem(3) has at least one energy solution in a cylinder QT with some T > 0. This solution can becontinued to the maximal interval [0, T∗],

T∗ = supT > 0 : ‖u(t)‖∞,Ω <∞ ∀ t < T S. Antontsev, S. Shmarev PDEs with Variable Nonlinearity 11

Sketch of proof: adaptation of Galerkin’s method

C∞0 (Ω) ⊂ Hs0(Ω) ⊂ Vt(Ω) (dense embeddings)

u(k) =∑ki=1 u

(k)i (t)ψi(x), ψi dense in W1,p

+

0 (Ω), (ψi,ψj)2,Ω = δij

monotonicity+

compactness+

integrationby parts in t

⇒ ∫

Q

[(ut + f)φ+

∑i

(|Diu|

pi(z)−2Diu)·Diφ

]dz = 0

integration by parts: finite differencesf(·, t+ h) − f(·, t)

h→ ∂tf wouldn’t work

f(x, t+ h) ∈ Vt+h(Ω), f(x, t) ∈ Vt(Ω), Vt(Ω) 6= Vt+h(Ω)

Lemma (Integration by parts)v,w ∈W(Q), vt,wt ∈W ′(Q), pi(z) are Log-continuous ⇒

∀a.e. t1, t2 ∈ (0, T ]∫t2t1

∫Ωvwt dz+

∫t2t1

∫Ωvtwdz =

∫Ωvwdx

∣∣∣t=t2t=t1

.


Localization properties in the isotropic case

Isotropic p-Laplacian, p = const: the alternative

ut = div(|∇u|p−2∇u) in Qu(x, 0) = u0(x) in Ω, u = 0 on Γp = const ∈ (1, 2)∪ (2,∞)

Counterpart localization properties

Slow diffusion ⇒ space localizationp > 2 ⇒ u0(x0) = 0 ⇒ ∃ t0 : u(x0, t) = 0 ∀ t ∈ [0, t0]

Fast diffusion ⇒ time localizationp ∈ (1, 2) ⇒ u(x, t) ≡ 0 for t > T∗, x ∈ Ω

No longer true if the diffusion is anisotropic or variable!


Anisotropic diffusion: new issues in localization

Dirichlet problem for the anisotropic p(z)-Laplacian ut =∑i

Di

(|Diu|

pi(z)−2Diu)+ f(z,u) in Q

u(x, 0) = u0 in Ω u = 0 on Γ

Localization (vanishing) properties

traditional technique: comparison

no explicit super-solutions ⇒ analysis of local energy functions

directional (anisotropic) localization in space

time localization in equations of fast-slow diffusion

localized solutions of eventually linear equations

S. Antontsev, J. I. Diaz, S. Shmarev “Energy methods for free boundary problems.Applications to Nonlinear PDEs and Continuum Mechanics", Progress in Nonlinear PDEs,v.48, Birkhauser (2002)


Finite speed of propagation

1 Finite speed of propagation of disturbances from the initial data

ut =∑i

Di

(|Diu|

pi(z)−2Diu)+ f pi(z) > p

− > 2

Finite speed of propagation. pi(z) > 2 - slow diffusionu0 = 0, f = 0 en Bρ0(x0) ⇒ u(x, t) = 0 en Bρ(t)(x0), ρ(t) = ρ0 − Ctγ

The set x ∈ Bρ(t)(x0) : u(x, t) = 0 shrinks as t grows


The waiting time effect

1 Finite speed of propagation of disturbances from the initial data2 The waiting time effect.

ut =∑i

Di

(|Diu|

pi(z)−2Diu)+ f pi(z) > p

− > 2

The waiting time effect. pi(z) > 2 - slow diffusionu0 = 0, f = 0 en Bρ(x0) ⇒ u(x, t) = 0 en Bρ(x0)× [0, t∗]

y ∈ ∂x ∈ Bρ0(x0) : u(x, t) = 0 is immobile on [0, t∗(y)]


The local energy methodReduction to a nonlinear ODI for the energy function

E(r) =

∫t0

∫Br(x0)

∑i

|Diu|p− dz

E ′(r) =

∫t0

∫∂Br(x0)

∑i

|Diu|p− dSdt for a.e. r ∈ (0, dist (x0,∂Ω))

ut =∑i

Di(|Diu|pi(z)−2Diu) + f(z) pi(x, t) ∈ (p−,p+)

u0 ≡ 0 in BR(x0). Multiply by u(z) and integrate by parts in Br(x0)× (0, t):

1

2‖u‖22,Br(t) +min

Ep+

p− (r), E(r)6

t∫0

∫∂Br

|u|∑i

|Diu|pi(z)−1 + F

trace-interpolation inequality + small oscillation of pi(z)⇒∀a.e. r ∈ (0,R) E(r) 6 C(t) (E ′)

1α (r) + F(r) α ∈ (0, 1) if p− > 2


Analysis of the ODI for the energy function

ut =∑i

Di

(|Diu|

pi(z)−2Diu)+ f

E(r) =

∫t0

∫Br(x0)

∑i

|Diu|p− p− > 2

⇒

Eα(t) 6 C(t)E ′(r) +Φ(r)

E ′(r) > 0 0 6 E(r) 6M

α ∈ (0, 1)

if Φ = 0,then

0 6 E1−α(r) 6M1−α −r− ρ0C(t)︸︷︷︸→ 0 as r grows

Big domain+

small total energy M =∑i

∫Q|Diu|

p− dz

⇒

localization:

u(x, t) ≡ 0 in Bρ(t)(x0)× [0, t∗]


Anisotropic diffusion: the choice of the energy function

ut =∑iDi

(|Diu|

pi(z)−2Diu)+ f(z), x = (x1, x ′)

E(s) =

t∫0

∫Ω∩ x1 > s

∑i

|Diu|pi(z) dz

E ′(s) = −

t∫0

∫Ω∩ x1 = s

∑i

|Diu|pi(z) dx ′ dt ∀a.e. s


Directional localization•

ut =∑iDi

(|Diu|

pi(z)−2Diu)+ f(z) pi(z) ∈ (p−,p+) ⊂ (1,∞)

• Multiply by u and integrate by parts in Ω∩ x1 > s× (0, t)

12‖u‖

22,Ω∩x1>s

+ E(r) 6∫t0

∫Ω∩x1=s

|u|∑i

|Diu|pi(z)−1 + F

embedding theorems in anisotropic spaces on Ω∩ x1 = s

• Eβ(s) + C(t)E ′(s) 6 Φ(s)

0 6 E(s) 6M, E ′(s) 6 0

1

p−1<

1

n− 1

n∑i=2

1

p+i6

1

n− 1⇒ β ∈ (0, 1) E(s) ≡M for x1 > s∗

Localization in the direction x1The fast diffusion equation admits localized in space solutions


Anisotropic diffusion: infinite waiting time

ut =∑i

Di

(|Diu|

pi−2Diu)+ f(z) pi(z) > 1

Eβ(s) + CE ′(s) 6 Φ(s) ∼ ‖u0‖22,Ω∩x1>s + ‖f‖22,(Ω∩x1>s)×(0,t)∫s0

s

Φ(z)

(z− s0)1/(1−β)dz <∞ 1

p−1<

1

n− 1

n∑i=2

1

p+i6

1

n− 1

Anisotropic diffusion ⇒ nonpropogation of disturbances !

y

x

u(x,t)=0

a

b

l x*

supp fsupp u0

0

t<T*


Nonpropagation of disturbances: a simple example

ut = uxx +(|uy|

q−2uy)y

q > 2

Separation of variables: u = X(x, t) · Y(y)

Xt

Xq−1−

Xxx

Xq−1︸︷︷︸const

=(|Yy|

q−2Yy)yY︸︷︷︸

constXt = Xxx + µXq−1 in (a,b)× (0, T)(|Y ′|q−2Y ′) ′ = µY in (c,d)µ = const > 0

Y(y) = C (L− y)α+ α =q+ 1

q− 1C = C(µ,q)


Extinction in a finite time

ut =

∑iDi

(|Diu|

pi(z)−2Diu)− c |u|σ(z)−2u+ f(z)

u = 0 on ΓT , u(x, 0) = u0(x) in Ω

Theorem (Diffusion/absorption balance)If f ≡ 0 for t > tf and

either c(z,u) > c0 > 0 and

1 <1

n

n∑i=1

1

p+i (t)+

1

σ+(t)61

n

n∑i=1

1

p−i (t)+

1

σ−(t)6 1+

2

n,

or c(z,u) > 0, and

1 <2

n

n∑i=1

1

p+i (t)62

n

n∑i=1

1

p−i (t)6 1+

2

n,

then every energy solution vanishes at a finite moment tu > tf.


Reduction to nonlinear ODI for the energy function

The energy function E(t) =

∫Ωu2(x, t)dx

ut =∑i

Di

(|Diu|

pi(z)−2Diu)− c |u|σ(z)−2u+ f(z)

Multiply by u and integrate over Ω

1

2E ′(t) +

∫Ω

(∑i

|Diu|pi(z) + c|u|σ(z)

)6∫Ω

|f| |u|

embedding theorems in anisotropic spaces ⇒E ′(t) + CEβ(t) 6 F(t) = K

∫Ωf2

1

β(t)=1

n

n∑i=1

1

p+i (t)+

1

σ+(t)or

1

β(t)=2

n

n∑i=1

1

p+i (t)

β(t) ∈ (0, 1), F(t) ≡ 0 for t > tf ⇒ E ≡ 0 for t > tu

Fast diffusion in one direction may cause extinction!


Conditions of extinction. Model equation: n = 2ut = Dx

(|Dxu|

p1−2Dxu)+Dy

(|Dyu|

p2−2Dyu)

p1,p2 = const

Anisotropic diffusion Isotropic diffusion


Equations which become linear as t→ ∞ut =

∑i

Di

(|Diu|

pi(z)−2Diu)− |u|σ(z)−2u,

supΩ pi(z)→ 2, supΩ σ(z)→ 2 as t→∞E(t) = ‖u(·, t)‖22,Ω :

E ′(t) + CEβ(t)(t) 6 0

1

β(t)=1

n

n∑i=1

1

p+i (t)+

1

σ+(t)→ 1 as t→∞

Theorem (Extinction in eventually linear equations)β(t) is monotone increasing

Eqn. C∫ t∗0‖u0‖

2(β(t)−1)2,Ω dt =

∫∞0

ds

es(1−β(s))has a root ⇒ u(x, t) ≡ 0 for t > t∗

Examples

1 ut = ∆u− |u|σ(t)−2u σ(t)→ 2 :

∫R

ds

es(σ(s)−1)<∞

2 ut =(|ux|

p(t)−2ux

)x

p(t)→ 2 :

∫R

ds

es(2−p(s))/2<∞


Space/time localization: an example

ut =

(|ux|

p−2ux)x+(|uy|

q−2uy)y

in QT = (0,a)× (0,b)× (0, T),

u = 0 on Γ , u(x,y, 0) = u0.

1

p+1

q> 1 ⇒

u(x,y, t) vanishes in a finite timeu(x, t) ≡ 0 ∀ t > T∗

1 < q < p ⇒ u(x,y, t) is localized in the direction x:u0 = 0 for x > x0 ⇒ u ≡ 0 for x > x ′


Blow-up: Kaplan’s method for semilinear equations

ut = ∆u+ b(z) uσ(z) − 1

u|Γ = 0, u(x, 0) = u0(x) > 0 in Ω

−∆φ = λφ in Ω,

φ = 0 on ∂Ω,∫Ωφdx = 1

DefinitionWe say that u(x, t) blows-up in a finite time if ∃ t∗ : ‖u(·, t)‖∞,Ω →∞ as t→ t∗.

A sufficient condition of blow-up:

µ(t) =

∫Ωu(x, t)φ(x)dx 6 ‖u‖∞,Ω

∫Ωφ(x)dx = ‖u(·, t)‖∞,Ω →∞ as t→ t∗


Blow-up: Kaplan’s method for semilinear equations

ut = ∆u+ b(z)︸︷︷︸>0

uσ(z) − 1

−∆φ = λφ in Ω, φ = 0 on ∂Ω

µ ′(t) > −λµ(t) + α(t)µσ−−1(t) − β(t), µ(0) > 0, µ(t) =

∫Ωu(z)ψ(x)dx

0 < b− 6 α(t) =

(∫Ωb

12−σ− (z)φ(x)dx

)2−σ−0 6 β(t) =

∫Ωb(z)φ(x)dx 6 b+ <∞

A− = mint>0

(α(t) −

λσ−(t)−1

σ−(t) − 1

), B+ = max

t>0

(β(t) +

σ−(t) − 2

σ−(t) − 1

)<∞.

Theorem (Nonexistence of global solutions)

Every weak solution blows-up if A− > 0, B+ <∞, A−µσ−(t)−1(0) > B+ for all t > 0 and

either σ−(t) = minx∈Ω σ(x, t) > σ− > 2, or

µ(0) > 1, σ−(t) > 2, σ−(t) 2 as t→∞,∫∞lnµ(0)

es(2−σ−(s))ds <∞


ExtensionsAn example:

ut = ∆u+ u1+ε(t)∫∞lnµ(0)

dτ

eτε(τ)<∞ ε(τ) = α

ln ττ

α > 1

Regional blow-up

µ(t) =

∫Du(x, t)φ(x)dx

−∆φ = λφ in D ⊂ Ω ,φ = 0 on ∂D

Equations with nonlocal reaction terms

ut = ∆u+ b(z)uσi(z)−1 + c(z)

∫Ωd(z)uσ2(z)−1 dx

interaction between b(z) and σ(z): every solution of the equation

ut = ∆u+ b(z)uσ(z)−1

blows-up if I =

∫Ω

ds

bqγ1−q (s, t)

<∞ with some q > 1σ−−1

, γ > 1.

For example:

|Ω| = |x| < 1, b = |x|−α ⇒ I =

∫10sn−1−α qγ

1−q ds <∞ ⇒ α <n(1− q)

γq


Blow-up in solutions of quasilinear equations ut = div

(a(z)|∇u|

p(x) −2∇u

)+ b(z)|u|

σ(x) −2

u(x, 0) = u0(x) in Ω, u = 0 on ΓT

Theorem1) σ− > 2, p+ = supΩ p(x) 6 σ− = infΩ σ(x), at(x, t) 6 0, bt(x, t) > 0 ,

2)

∫T0

(maxx∈Ω

|at(x, t)|+ |bt(x, t)|)dt <∞, |u0|

σ(x) ∈ L1(Ω), |∇u0|p(x) ∈ L1(Ω)

3)

∫Ω

(a(x, 0)p(x)

|∇u0|p(x) −b(x, 0)σ(x)

|u0|σ(x)

)dx 6 0

⇒ every nonstationary solution blows-up in a finite time:

∃ t∗ ≡ t∗(Ω,‖u0‖∞) <∞ : ‖u(·, t)‖∞,Ω →∞ as t→ t∗.

Reduction to the 2nd-order differential inequality for f(t) =1

2

∫t0

∫Ωu2(z)dx :

1 p+ = supΩp(x) > 2 ⇒ f(t) 6 2λf ′′(t)

1

σ−6 λ 6

1

p+,

2 p+ ∈ (1, 2], σ− > p+ ⇒ K (f ′(t))σ−

2 6 f ′′(t) K =

(λ−

1

σ−

)b−

λ


Wave equation with p(x, t)-Laplacian

utt = div

(|∇u|p(z)−2∇u+ ε∇ut

)+ b|u|σ(z)−2u+ f

u(x, 0) = u0(x), ut(x, 0) = u1(x), x ∈ Ω,u|ΓT = 0, ΓT = ∂Ω× (0, T), ε = const > 0.

(4)

The main function space:

W =

u : u,∇ut ∈ L2 (QT ) , ut, |∇u|

p(z)2 , |u|

σ(z)2 ∈ L∞ (0, T ;L2 (Ω)

), u|ΓT = 0

‖u‖W = ‖u‖2,QT + ‖u‖σ(·),QT + ‖ut‖L∞(0,T ; L2(Ω))

+‖∇u‖L∞(0,T ; L2(Ω)) + ‖∇ut‖2,QT + ‖∇u‖p(·),QT

Definition (The energy solution u ∈ W)

1 u(·, t) u0 in W1,20 (Ω)∩W1,p(·,0)(Ω), ut(·, t) u1 in L2(Ω)

2 ∀ϕ ∈ C∞ (0, T ;C∞0 (Ω)

)such that ϕ(x, T) = 0

∫QT

(−utϕt +

(|∇u|p(·)−2∇u+ ε∇ut

)· ∇ϕ− b |u|σ(·)−2 uϕ

)dxdt

=

∫Ωu1ϕ(·, 0)dx+

∫QT

fϕdx


Local and global in time existence

utt = div

(|∇u|p(z)−2∇u+ ε∇ut

)+ b|u|σ(z)−2u+ f


(5)

u0 ∈ L2(Ω)∩W1,20 (Ω)∩W1,p(·,0)(Ω), u1 ∈ L2(Ω), f ∈ L2(QT )p(z) ∈ Clog(QT ) 1 < p− 6 p(x, t) 6 p+ <∞, |pt| = −pt 6 Cp ,

1 < σ− 6 σ(x, t) 6 σ+ 6∞, 0 6 σt 6 Cσ

(6)

Theorem (Global in time existence)

Let conditions (6) be fulfilled and one of the following conditions holds: a) b < 0 ,

b) σ+ 6 2, or σ+ < p− . Then problem (5) has at least one global in time energy solution.

Theorem (Local in time existence)

Let conditions (6) be fulfilled and b > 0, 2 < σ− 6 σ+ <n+ 2

np−,

2n

n+ 2< p− . Then

problem (5) has at least one solution on a small time interval [0, T0).


The energy relations

The energy functional: E(t) =

∫Ω

[|ut(·, t)|2

2+

|∇u|p(·,t)

p(·, t)− b

|u|σ(·,t)

σ(·, t)

]dx

E ′(t) + ε

∫Ω

|∇ut(·, t)|2 dx = Λ1 +Λ2 +Λ3

Λ1 =

∫Ω

[−

|∇u|p

p2(1− p ln |∇u|) pt

]dx,

Λ2 =

∫Ω

(b |u|σ

σ2(1− σ ln |u|) σt

)dx, Λ3 =

∫Ωfutdx.

1 pt = σt = f = 0 ⇒ E(t) + ε

∫t0

∫Ω

|∇ut(x, s)|2 dxds = E(0) ∀t > 0

2 f = 0 ⇒ E(t) + ε

∫t0

∫Ω

|∇ut(x, s)|2 dxds 6 E(0) + Ct

with the constant C = ( bσ2−Cσ + 1

p2−Cp)|Ω|.


Conditions of the finite time blow-up

utt = div

(|∇u|p(z)−2∇u+ ε∇ut

)+ b|u|σ(z)−2u+ f


(7)

Theorem

If b < 0, 2 < p− 6 p+ < σ−, E0 < 0, Z(0) = 2(u0,u1)2,Ω + ε‖∇u0‖22,Ω > 0 , then

∃tmax <∞ :

∫Ω

|u|σdx > CZ(0)

(1−

t(µ− 1)

C(Z(0))µ−1

)− 1µ−1 →∞ as t→ tmax,

where C > 0 and µ > 1 are constants depending only on the data.

Weak dependence of the exponents p(z), σ(z) on t (the constants Cp,Cσ are small):

δ = max (Cp,Cσ) 6 |E(0)| (tmax

(1

p2−+

b

σ2−

)|Ω|)−1


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Bibliography

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S. Antontsev, S. Shmarev. "Uniqueness and comparison theorems for solutions of doubly nonlinearparabolic equations with nonstandard growth conditions". CPAA, 12 (4) (2013), pp. 1827–1846.

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S. Antontsev. "Wave equation with p(x, t)-Laplacian and damping term: Blow-up of solutions".C.R. Mecanique, 339 (2011), pp. 751–755.

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Thank you!S. Antontsev, S. Shmarev PDEs with Variable Nonlinearity 37

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