# Global Attractor for Doubly Nonlinear Parabolic Equation

Authors: Yongjun Li; Suyun Wang; Yanhong Zhang
Journal: ISRN Applied Mathematics
 (2012)

---

## Abstract

Our aim in this paper is to study the long-time behavior for a class of doubly nonlinear parabolic equations. First we show that the problem has a unique solution. Then we prove that the semigroup corresponding to the problem is norm-to-weak continuous inLq and H01. Finally we establish the existence of global attractor of the problem in Lq and H01.


---

## Body

## 1. Introduction

We study the long-time behavior (in terms of attractors) of the solution of the following problem:(1.1)∂α(u)∂t-Δu+f(u)=g(x),x∈Ω,u(x,t)|∂Ω=0,u(x,0)=u0(x).
Here Ω is a bounded smooth domain in ℝN(N≥1) and g(x) is a given function in L2(Ω).Such equations appear, for example, in the study of gas filtration (so-called porous medium equation [1]). It has been extensively studied when α is linear [1–5], and the existence of attractors has been proved in [6–11] (for α=I).Our aim in this paper is to extend the result of [1–4] to the more general equation (1.1). We make the following assumptions:
(1.2)α(s)∈C1(R),α(0)=0,α′(s)≥α0,α0>0,s∈R,(1.3)α1|s|r+2-α3≤α(s)s≤α2|s|r+2+α3,α1,α2>0,α3≥0,r≥0,(1.4)α′(s)≤α4|s|r+α5,α4>0,α5≥0,s∈R,(1.5)f(s)∈C(R),γ1|s|q-γ3≤f(s)s≤γ2|s|q+γ3,s∈R,γ1,γ2>0,γ3≥0,q≥r+2.
There exists a constant C0≥0, such that
(1.6)C0α(s)+f(s)is increasing.By hypotheses (1.2)–(1.5), α and f are nonlinear functions with polynomial growth of arbitrary order. Here α is more general than in [1–4](where α is linear growth), which is an essential difficulty in proving the existence of global attractor. To problem (1.1), the key points are to obtain the continuous and compactness of semigroup. By using Legendre transform and the asymptotic a priori estimate method introduced in [10, 11], we show the existence of global attractor.This paper is organized as follows. In Section2, we recall some basic concepts about the global attractor. In Section 3, we show the uniqueness of solution and norm-to-weak continuous semigroup for (1.1). In Section 4, we verify the asymptotic compactness of the semigroup S(t) in Lq(Ω) and prove the existence of the (Lr+2(Ω),Lq(Ω))-global attractor under the hypotheses (1.2)–(1.6). Finally, in Section 5, we prove the existence of the (Lr+2(Ω),H01(Ω))-global attractor for N≤2.Throughout this paper we use the following notation:H=L2(Ω), and the norms in H01(Ω) and Lp(Ω)(1≤p≤∞) are denoted by ∥u∥2=∫Ω‍|∇u|2dx and |u|pp=∫Ω‍|u|pdx, respectively; Ω(u≥M)={x∈Ω:u(x)≥M} and Ω(u≤-M)={x∈Ω:u(x)≤-M}; m(Ω) or |Ω| denotes Lebesgue measure of Ω; sometimes for special differentiation, we denote the different positive constants by c1, c2,….


## 2. Preliminaries

In this section, we recall some basic concepts about the global attractors.Definition 2.1 (see [6–8]).
Let{S(t)}t≥0 be a semigroup on a Banach space X. A subset 𝒜⊂X is called a global attractor for the semigroup if 𝒜 is compact in X and enjoys the following properties:(1)
𝒜 is an invariant set, that is, S(t)𝒜=𝒜 for any t≥0;(2)
𝒜 attracts all bounded sets of X, that is, for any bounded subset B of X(2.1)dist⁡(S(t)B,A)⟶0,ast⟶∞,where dist(A,B) is the Hausdorff semidistance of the two sets A and B:
(2.2)dist⁡(A,B)=sup⁡x∈Ainf⁡y∈B‖x-y‖X.
And a subsetB0 of X is called a bounded absorbing set of the semigroup {S(t)}t≥0 in X, if for any bounded B of X, there exists some TB≥0, such that S(t)B⊂B0 for any t≥t0.Definition 2.2 (see [11]).
LetX be a Banach space and let {S(t)}t≥0 be a family of operator in X. We say that {S(t)}t≥0 is norm-to-weak continuous semigroup in X, if {S(t)}t≥0 satisfies(1)
S(0)=I (the identity);(2)
S(t)S(s)=S(t+s),∀t,s≥0;(3)
S(tn)xn⇀S(x)x if tn→t and xn→x in X.Definition 2.3 (see [11]).
A set𝒜⊂X, which is invariant, closed in X, compact in Z and attracts the bounded subsets of X in the topology of Z, is called an (X,Z)-global attractor.Definition 2.4 (see [11]).
Let{S(t)}t≥0 be a semigroup on Banach space X. {S(t)}t≥0 is called (X,Z)-asymptotically compact, if for any bounded (inX) sequence{xn}n=1∞⊂X and tn≥0, tn→∞ as n→∞, {S(tn)xn}n=1∞ has a convergence subsequence with respect to the topology of Z.Definition 2.5 (see [11]).
Let{S(t)}t≥0 be a semigroup on Banach space X. A bounded subset B0 of Z is called an (X,Z)-bounded absorbing set, if for any bounded set B of X, there exists some tB≥0, such that S(t)B⊂B0 for any t≥tB.Theorem 2.6 (see [11]).
LetX be a Banach space and let {S(t)}t≥0 be a norm-to-weak continuous semigroup on X. Then {S(t)}t≥0 has a global attractor in X provided that the following conditions hold:(1)
{S(t)}t≥0 has a bounded absorbing set B0 in X;(2)
{S(t)}t≥0 is asymptotically compact in X.Theorem 2.7 (see [11]).
Let{S(t)}t≥0 be a norm-to-weak continuous semigroup on X. One assumes that S(t)X⊂Lp(Ω). Then {S(t)}t≥0 has an (X,Lp(Ω))-global attractor provided that the following conditions hold:(1)
{S(t)}t≥0 has an (X,Lp(Ω))-bounded absorbing set B0⊂Lp(Ω);(2)
there is aq(1≤q≤p) such that {S(t)}t≥0 is (X,Lq(Ω))-asymptotically compact;(3)
for anyɛ>0 and for any bounded B⊂X, there exist constants M=M(ɛ,B) and T=T(ɛ,B), such that
(2.3)∫Ω(|S(t)u0|≥M)‍|S(t)u0|pdx<ɛfor  anyu0∈Bandt≥T.Lemma 2.8 (see [11]).
Let{S(t)}t≥0 be a semigroup on Lp(Ω)(p≥1) and suppose that {S(t)}t≥0 has a bounded absorbing set in Lp(Ω). Then for any ɛ>0 and any bounded subset B⊂Lp(Ω), there exist positive constants T=T(B) and M=M(ɛ) such that
(2.4)m(Ω(|S(t)u0|≥M))≤Cɛfor  anyu0∈Bandt≥T,
where the positive constant C is independent of B, T, and ɛ.


## 3. Uniqueness of Solution and Norm-to-Weak Continuous Semigroup

The existence of weak solution for (1.1) can be obtained by the standard Faedo-Galerkin approximation method. Here we only state the result.Lemma 3.1.
Assume thatg(x)∈L2(Ω), α and f satisfying (1.3)–(1.5), u0(x)∈Lr+2(Ω). Then for any initial data u0(x)∈Lr+2(Ω), there exists solution u(x,t) for (1.1) which satisfies
(3.1)u(x,t)∈C(0,T;L2(Ω))∩L2(0,T;H01(Ω))∩Lq(0,T;Lq(Ω)).We now show that the solution is uniqueness and continuous dependent on initial conditions.Theorem 3.2.
Assume thatg(x)∈L2(Ω),u0(x)∈Lr+2(Ω), α and f satisfying (1.3)–(1.6). Then there exists a unique solution of (1.1).Proof.
Suppose thatu(t),v(t) be two solutions of (1.1) with initial conditions u0(x), v0(x); then
(3.2)∂(α(u)-α(v))∂t-Δ(u-v)+f(u)-f(v)=0,
that is
(3.3)∂(α(u)-α(v))∂t-Δ(u-v)+(C0α(u)+f(u))-(C0α(v)+f(v))=C0(α(u)-α(v)).
We define the sign function by
(3.4)sign⁡(τ)={1ifτ>0,0ifτ=0,-1ifτ<0.
Multiplying (3.3) by sign(u-v) and integrating in Ω, we obtain
(3.5)ddt∫Ω‍|α(u)-α(v)|dx-∫Ω‍Δ(u-v)sign⁡(u-v)ddt∫Ω‍+∫Ω‍[(C0α(u)+f(u))-(C0α(v)+f(v))]sign⁡(α(u)-α(v))dxddt=C0∫Ω‍|α(u)-α(v)|dx.
Using  (1.6), we get
(3.6)∫Ω‍[(C0α(u)+f(u))-(C0α(v)+f(v))]sign⁡(α(u)-α(v))dx≥0.
Since sign(u-v)=limɛ→0+(u-v)/(ɛ+|u-v|), by dominated convergence theorem, we have
(3.7)-∫Ω‍Δ(u-v)sign⁡(u-v)dx=-lim⁡ɛ→0+∫Ω‍Δ(u-v)u-vɛ+|u-v|dx∫Ω‍=lim⁡ɛ→0+∫Ω‍∇(u-v)∇(u-vɛ+|u-v|)dx≥0.
So
(3.8)ddt∫Ω‍|α(u)-α(v)|dx≤C0∫Ω‍|α(u)-α(v)|dx.
By Gronwall inequality, we get
(3.9)∫Ω‍|α(u(t))-α(v(t))|dx≤∫Ω‍|α(u0)-α(v0)|dx.
From (1.2), we have
(3.10)∫Ω‍|u(t)-v(t)|dx≤1α0∫Ω‍|α(u0)-α(v0)|dx,
which gives continuous dependence on initial conditions and uniqueness of solution in L1(Ω).
By Theorem3.2, we can define the operator semigroup {S(t)}t≥0 in L1(Ω) as the following:
(3.11)S(t)u0:R+×Lr+2(Ω)⟶L1(Ω),
which is continuous in L1(Ω).
Sinceα is a continuous increasing function with α(0)=0, we define for t∈ℝ,
(3.12)ψ(t)=∫0t‍α(τ)dτ.
Then the Legendre transform ψ* is defined by
(3.13)ψ*(τ)=sup⁡s∈R{τs-ψ(s)}.
Note that
(3.14)ψ*(τ)≥0,ψ*(α(τ))+ψ(τ)=τα(τ),ψ*(α(τ))≤τα(τ).Theorem 3.3.
Assume that the conditions (1.2)–(1.6) are satisfied, g(x)∈L2(Ω). Then the semigroup {S(t)}t≥0 is norm-to-weak continuous in Lq(Ω) and H01(Ω).Proof.
Letum0(x)→u0(x) in Lr+2(Ω), um(t),u(t) be the solutions of (1.1) corresponding to initial date um0(x), u0(x). In (1.1), replace u(t) by um(t). Multiplying (1.1) by um(t) and integrating in Ω, we get
(3.15)ddt∫Ω‍ψ*(α(um(t)))dx+|∇um|22+(f(um),um)=(g,um).
Applying Young inequality, we have
(3.16)|∫Ω‍g(x)umdx|≤γ12|um|qq+c∫Ω‍|g(x)|q′dx.(1q+1q′=1).
So
(3.17)ddt∫Ω‍ψ*(α(um(t)))dx+|∇um|22+γ12|um|qq≤γ3|Ω|+c∫Ω‍|g|q′dx.
Integrating from 0 to T, we obtain
(3.18)∫Ω‍ψ*(α(um(T)))dx+∫0T‍|∇um|22dt+γ12∫0T‍|um|qqdtdx+≤∫Ω‍ψ*(α(um0))dx+cTdx+≤∫Ω‍um0α(um0)dx+α3|Ω|+cTdx+≤α2|um0|r+2r+2+cT+α3|Ω|.um0→u0 in Lr+2(Ω), so there existence M>0, such that |um0|r+2r+2≤M. {um(t)} Is bounded in L2(0,T;H01(Ω)) and Lq(0,T;Lq(Ω)). Therefore, there exists weak convergent subsequence {umk(t)} of {um(t)} in L2(0,T;H01(Ω)) and Lq(0,T;Lq(Ω). Let um(t)⇀v(t) obviously, v(t) be a solution of (1.1) satisfying the initial value condition v(0)=u0(x). By the unique of solution for (1.1), we have u(t)=v(t), that is, umk⇀u(t) in L2(0,T;H01(Ω)) and Lq(0,T;Lq(Ω)). By Definition 2.2, Theorem 3.3 holds.Remark 3.4.
The semigroup{S(t)}t≥0 is norm-to-weak continuous in L2(Ω).


## 4.(Lr+2(Ω),Lq(Ω))-Global Attractor

By Theorem3.3, we can define operator semigroup {S(t)}t≥0 as the following:
(4.1)S(t)u0:R+×Lr+2(Ω)⟶Lq(Ω),S(t)u0:R+×Lr+2(Ω)⟶H01(Ω).Theorem 4.1.
Assume that the conditions (1.2)–(1.6) are satisfied, g(x)∈L2(Ω). Then the semigroup {S(t)}t≥0 exists bounded absorbing sets in Lq(Ω) and H01(Ω); that is, for arbitrary bounded set B⊂Lr+2(Ω), there exist M>0, T(B)>0, ∀t≥T, ∀u0(x)∈B. We have
(4.2)|u(t)|qq+|∇u|22≤M.Proof.
Letu(t) be the solution of (1.1) with initial date u0(x); taking scalar product with u(t) in (1.1), we deduce that
(4.3)ddt∫Ω‍ψ*(α(u))dx+|∇u|22+(f(u),u)=(g,u).
By (1.5), we obtain
(4.4)ddt∫Ω‍ψ*(α(u))dx+|∇u|22+γ1|u|qq-γ3|Ω|≤(g,u).
Using Young inequality, we get
(4.5)|(g,u)|≤γ12∫Ω‍|u|qdx+c1∫Ω‍|g|q′dx.
By (4.4), we have
(4.6)ddt∫Ω‍ψ*(α(u))dx+|∇u|22+γ12|u|qq≤γ3|Ω|+c1∫Ω‍|g|q′dx.
Since q≥r+2, by (1.3), there exist c2,c3>0 such that
(4.7)γ14∫Ω‍|u|qdx≥c3∫Ω‍ψ*(α(u))dx-c2.
Hence,
(4.8)ddt∫Ω‍ψ*(α(u))dx+|∇u|22+γ14|u|qq+c3∫Ω‍ψ*(α(u))dx≤c4.
We get
(4.9)ddt∫Ω‍ψ*(α(u))dx+c3∫Ω‍ψ*(α(u))dx≤c4.
By the Gronwall lemma, we have
(4.10)∫Ω‍ψ*(α(u(t)))≤e-c3t∫Ω‍ψ*(α(u0))dx+c4c3.
Therefore,
(4.11)∫Ω‍ψ*(α(u0))dx≤∫Ω‍u0α(u0)dx.
We obtain
(4.12)∫Ω‍ψ*(α(u(t)))≤α2|u0|r+2r+2e-c3t+c4c3+α3|Ω|≤c5.
It follows from (4.8) that there exists T1>0,∀t≥T1; we get
(4.13)∫tt+1‍|∇u|22ds+∫tt+1‍|u|qqds≤c6.
Multiplying (1.1) by ut and integrating over Ω, we get
(4.14)∫Ω‍α′(u)ut2dx+ddt∫Ω‍(|∇u|2+F(u))dx=(g,ut)≤12∫Ω‍α′(u)ut2dx+12α0|g(x)|22,
where F(u)=∫0u‍f(s)ds. Now by (1.2) we get
(4.15)ddt∫Ω‍(|∇u|2+F(u))dx≤12α0|g(x)|22.
It follows from (1.5) that there exist γ1′,γ2′>0,γ3′≥0 such that
(4.16)γ1′|s|q-γ3′≤F(s)≤γ2′|s|q+γ3′.
Using the uniform Gronwall Lemma that there exists T2>0,∀t≥T2, we have
(4.17)∫Ω‍(|∇u|2+F(u))dx≤c7.
By (4.16), ∀t≥T=max{T1,T2}, we obtain
(4.18)|∇u|22+|u|qq≤c8.
Therefor, the semigroup {S(t)} exists bounded absorbing set in H01(Ω) and Lq(Ω); it follows from Theorem 2.6 that  we have the following.Theorem 4.2.
Assume that the conditions (1.2)–(1.6) are satisfied, g(x)∈L2(Ω). Then the semigroup {S(t)}t≥0 has a (Lr+2(Ω),L2(Ω))-global attractor, which is nonempty, compact, invariant in L2(Ω) and attracts every bounded subset of Lr+2(Ω) with respect to L2(Ω) norm.In the following, we will give the asymptotic a priori estimate of{S(t)}t≥0 with respect to Lq(Ω)-norm, which plays a crucial role in the proof of the (Lr+2(Ω),Lq(Ω))-global attractor.Lemma 4.3.
Assume that the conditions (1.2)–(1.6) are satisfied, g(x)∈L2(Ω), q≥max{2r,r+2}, ∀B⊂Lr+2(Ω). For any ɛ>0, there exist positive constants M(=M(ɛ)) and T(=T(ɛ)) such that
(4.19)∫Ω(|u(t)|≥M)‍|u(t)|qdx<cɛ,for  anyt≥Tandu0(x)∈B.Proof.
By (1.3) and (1.5), we find that there exists M1>0,∀|u|≥M1 such that
(4.20)f(u)u≥γ12|u|q,α12|u|r+1≤|α(u)|≤2α2|u|r+1.
LettingM2=max{1,α1/(2)|M1|r+1}, when |α(u)|≥M2, then |u|≥M1. Multiplying (1.1) with |(α(u)-M2)+|(q/(r+1))-2(α(u)-M2)+, we get
(4.21)r+1qddt∫Ω‍|(α(u)-M2)+|q/(r+1)dx+∫Ω‍∇u∇(|(α(u)-M2)+|q/(r+1)-2(α(u)-M2)+)dxr+1qddt+∫Ω‍f(u)|(α(u)-M2)+|q/(r+1)-2(α(u)-M2)+dxr+1q=∫Ω‍g(x)|(α(u)-M2)+|q/(r+1)-2(α(u)-M2)+dx,
where (α(u)-M2)+ denote the positive part of (α(u)-M2), that is:
(4.22)(α(u)-M2)+={α(u)-M2,α(u)≥M2,0,α(u)<M2.
Thus we have
(4.23)∫Ω‍∇u∇(|(α(u)-M2)+|q/(r+1)-2(α(u)-M2)+)dx∇u∇=∫Ω(α(u)≥M)‍∇u∇(|(α(u)-M2)|q/(r+1)-1)dx∇u∇=(qr+1-1)∫Ω(α(u)≥M)‍α′(u)|(α(u)-M2)|q/(r+1)-2|∇u|2dx≥0,∫Ω‍f(u)|(α(u)-M2)+|q/(r+1)-2(α(u)-M2)+dx∇u∇≥γ12∫Ω‍|u|q-1|(α(u)-M2)+|q/(r+1)-1dx∇u∇≥c∫Ω‍|α(u)|(q-1)/(r+1)|(α(u)-M2)+|q/(r+1)-1dx∇u∇≥c∫Ω‍|(α(u)-M2)+|(2q-r-2)/(r+1)dx,|∫Ω‍g(x)|(α(u)-M2)+|q/(r+1)-2(α(u)-M2)+dx|∇u∇≤∫Ω‍|g(x)||(α(u)-M2)+|q/(r+1)-1dx∇u∇≤c2∫Ω‍|(α(u)-M2)+|(2q-r-2)/(r+1)dx+c∫Ω(α(u)≥M2)‍|g(x)|(2q-r-2)/(q-1)dx.
We obtain
(4.24)ddt∫Ω‍|(α(u)-M2)+|q/(r+1)dx+c∫Ω‍|(α(u)-M2)+|(2q-r-2)/(r+1)dxddt≤c∫Ω(α(u)≥M2)‍|g(x)|(2q-r-2)/(q-1)dx.
Since q≥r+2, we have
(4.25)2q-r-2r+1≥qr+1,1<2q-r-2q-1≤2.
So
(4.26)ddt∫Ω‍|(α(u)-M2)+|q/(r+1)dx+c∫Ω‍|(α(u)-M2)+|q/(r+1)dxddt≤c∫Ω(α(u)≥M2)‍|g(x)|(2q-r-2)/(q-1)dx.
By Gronwall inequality, there exists T(B)>0,∀t≥T(B) such that
(4.27)∫Ω‍|(α(u(t))-M2)+|q/(r+1)dx≤e-c(t-T)∫Ω‍|(α(u(T))-M2)+|q/(r+1)dx∫Ω‍+c(∫Ω(α(u)≥M2)‍|g(x)|(2q-r-2)/(q-1))dx.
For anyɛ>0 there exists δ>0; for any e⊂Ω, m(e)<δ, we have
(4.28)∫e‍|g(x)|(2q-r-2)/(q-1)dx<ɛ.
LetingT(t)=α(u(t)), obviously {T(t)}t≥0 is a semigroup in L1(Ω). By (1.5) and (4.18), {T(t)}t≥0 has a bounded absorbing set in L1(Ω). Combining Lemma 2.8, for any ɛ>0, there exist M≥M2,T1≥T(B), for any t>T1, we have
(4.29)m(Ω(α(u)≥M))<min⁡{δ,ɛ}.
Hence
(4.30)∫Ω‍|(α(u(t))-M2)+|q/(r+1)dx<cɛ,
that is
(4.31)∫Ω(α(u)≥M)‍(|α(u(t))|-M)q/(r+1)dx<cɛ.
By repeating the same step above and multiplying (1.1) with |(α(u)+M2)-|q/(r+1)-2(α(u)+M2)-, we get
(4.32)∫Ω(α(u)≤-M)‍(|α(u(t))|-M)q/(r+1)dx<cɛ,
where
(4.33)(α(u)+M2)-={α(u)+M2,α(u)≤-M2,0,α(u)>-M2.
Combining (4.31) and (4.32), we have
(4.34)∫Ω(|α(u)|≥2M)‍|α(u(t))|q/(r+1)dx∫Ω‍=∫Ω(|α(u)|≥2M)‍(|α(u(t))|-M+M)q/(r+1)dx∫Ω‍≤c(∫Ω(|α(u)|≥2M)‍(|α(u(t))|-M)q/(r+1)dx+∫Ω(|α(u)|≥2M)‍Mq/(r+1)dx)∫Ω‍≤c(∫Ω(|α(u)|≥M)‍(|α(u(t))|-M)q/(r+1)dx+∫Ω(|α(u)|≥M)‍(|α(u(t))|-M)q/(r+1)dx)≤cɛ.
Thanks to (4.20), we have
(4.35)∫Ω(|u(t)|≥M)‍|u(t)|qdx<cɛ.
From Theorem4.2, we know that the semigroup {S(t)}t≥0 is asymptotically compact in L2(Ω); using Theorem 4.1, Lemma 4.3, and Theorem 2.7, we get the following.Theorem 4.4.
Assume that the conditions (1.2)–(1.6) are satisfied, g(x)∈L2(Ω). Then the semigroup {S(t)}t≥0 has a (Lr+2(Ω),Lq(Ω))-global attractor, which is nonempty, compact, invariant in Lq(Ω) and attracts every bounded subset of Lr+2(Ω) with respect to Lq(Ω) norm.


## 5.(Lr+2(Ω),H01(Ω))-Global Attractor

In this section, we want to prove the(Lr+2(Ω),H01(Ω))-global attractor. However, for general N-dimension space, our methods can not work, so we make the following assumption:
(5.1)N≤2,α(s)∈C2(R),f(s)∈C1(Ω),s∈R.Lemma 5.1.
Assume that the conditions (1.2), (1.3), (1.5), (1.6), and (5.1) are satisfied, g(x)∈L2(Ω). For any bounded set B⊂Lr+2(Ω), there exists t*(B)>0; for any t≥t*, we have
(5.2)∫Ω‍α′(u)|ut|2dx≤c.Proof.
Multiplying (1.1) by ut and integrating over Ω, we have
(5.3)∫Ω‍α′(u)|ut|2dx+ddt∫Ω‍(|∇u|2+F(u))dx=(g(x),ut)≤∫Ω‍α′(u)2|ut|2dx+|g(x)|222α0,
and then
(5.4)∫Ω‍α′(u)|ut|2+cddt∫Ω‍|∇u|2+F(u)dx≤c.
By (4.16) and (4.18), there exists T(B)>0 for any t≥T(B); we get
(5.5)∫Ω‍(|∇u|2+F(u))dx≤c.
Integrating (5.4) from t to t+1, we obtain
(5.6)∫tt+1‍∫Ω‍α′(u)|ut|2dxds≤c,t≥T(B).
By differentiating (1.1) in time t, we have
(5.7)α′′(u)ut2+α′(u)utt-Δut+f′(u)ut=0.
Letting r(t)=α′(u)ut2, we get
(5.8)ddt∫Ω‍r(t)dx=∫Ω‍α′′(u)ut3dx+2∫Ω‍α′(u)ututtdx,
that is
(5.9)12ddt∫Ω‍r(t)dx=12∫Ω‍α′′(u)ut3dx+∫Ω‍α′(u)ututtdx.
Multiplying (5.7) by ut and integrating over Ω, we get
(5.10)∫Ω‍α′′(u)ut3dx+∫Ω‍α′(u)uttutdx=-∫Ω‍|∇ut|2dx-∫Ω‍f′(u)ut2dx.
Hence from (5.9), we obtain
(5.11)12ddt∫Ω‍r(t)dx=-12∫Ω‍α′′(u)ut3dx-∫Ω‍|∇ut|2dx-∫Ω‍f′(u)ut2dx.
SinceN≤2, from Theorem 4.1 the semigroup {S(t)}t≥0 has a bounded absorbing set in H01(Ω). Applying Sobolev embedding theorem, the semigroup {S(t)}t≥0 is bounded in L∞(Ω); hence,
(5.12)12ddt∫Ω‍r(t)dx+∫Ω‍|∇ut|2dx≤M∫Ω‍|ut|3dx+c∫Ω‍|ut|2dx.
SinceN≤2, the imbedding H1/3(Ω)↪L3(Ω) is compact; hence
(5.13)|ut|33≤c|ut|H1/3(Ω)3.
Using interpolation inequality for Sobolev space, we get
(5.14)|ut|H1/3(Ω)3≤c|∇ut|2|ut|22≤12|∇ut|22+c|ut|24.
Therefore, from (5.12), we have
(5.15)12ddt∫Ω‍r(t)dx+∫Ω‍|∇ut|2dx≤c|ut|22+c|ut|24≤c|ut|24+c=c(∫Ω|ut|2dx)2+c≤c(∫Ω‍α′(u)|ut|2dx)2+c≤c(∫Ω‍r(t)dx)2+c.
Thanks to uniform Gronwall inequality, there exists T(B)>0 for any t≥T(B); we have
(5.16)∫Ω‍r(t)dx≤c.
Now we prove the semigroup{S(t)}t≥0 is asymptotically compact in H01(Ω).Lemma 5.2.
Assume that the conditions (1.2), (1.3), (1.5), (1.6) and (5.1) are satisfied, g(x)∈L2(Ω). Then the semigroup {S(t)}t≥0 is asymptotically compact in H01(Ω).Proof.
LetB is a bounded absorbing set in H01(Ω), xn∈B, tn→∞ as n→∞, un(t)=S(tn)xn. Now we will prove that there exists Cauchy sequence of {un(t)} in H01(Ω). By Theorem 4.2, we know that {S(t)}t≥0 exists global attractor in L2(Ω), therefore there exists  Cauchy sequence of {un(t)} in L2(Ω), we denote by {un(t)}. Hence, ∀0<ɛ<1,∃N,∀n,m>N:
(5.17)|un-um|22<ɛ2.
In view of (1.1), we get
(5.18)α′(um)umt-α′(un)unt-Δ(um-un)+f(um)-f(un)=0.
Multiplying the above equality by um-un and integrating over Ω, we obtain
(5.19)|∇um-∇un|22∇um-≤-∫Ω‍(α′(um)umt-α′(un)unt)(um-un)dx-∫Ω‍(f(um)-f(un))(um-un)dx∇um-≤c(∫Ω‍|umt||um-un|dx+∫Ω‍|unt||um-un|dx)-∫Ω‍f′(ξ)|um-un|2dx∇um-≤c(|umt|2|um-un|2+|unt|2|um-un|2)+c|um-un|22.
Hence, for any n,m>N, we get
(5.20)|∇um-∇un|22<cɛ.
That is to say, the semigroup {S(t)}t≥0 is asymptotically compact in H01(Ω).Thus from Theorem4.1 and Lemma 5.2, we have the following.Theorem 5.3.
Assume that the conditions (1.2), (1.3), (1.5), (1.6) and (5.1) are satisfied, g(x)∈L2(Ω). Then the semigroup {S(t)}t≥0 has a (Lr+2(Ω),H01(Ω))-global attractor, which is nonempty, compact, invariant in H01(Ω) and attracts every bounded subset of Lr+2(Ω) with respect to H01(Ω) norm.



---

*Source: 956291-2012-07-02.xml*