# Asymptotic Behavior of a Class of Degenerate Parabolic Equations

Authors: Hongtao Li; Shan Ma
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We investigate the asymptotic behavior of solutions of a class of degenerate parabolic equations in a bounded domainΩ⊂ℝn (n⩾2) with a polynomial growth nonlinearity of arbitrary order. The existence of global attractors is proved in L2(Ω), Lp(Ω), and H01,a(Ω), respectively, when H01,a(Ω) can be just compactly embedded into Lr(Ω) (r<2) but not L2(Ω).


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## Body

## 1. Introduction

Let us consider the following degenerate parabolic equations:(1.1)∂u∂t-div(a(x)∇u)+f(u)=g(x)inΩ×ℝ+,u=0on∂Ω×ℝ+,u(x,0)=u0inΩ,
where Ω is a bounded domain in ℝn (n⩾2), with smooth boundary ∂Ω, a is a given nonnegative function, and f is a 𝒞1 function satisfying
(1.2)f′(s)⩾-l,(1.3)C1|s|p-C0⩽f(s)s⩽C2|s|p+C0,p⩾2,
both for all s∈ℝ.For the long-time behavior problems of the classical evolutionary equations, especially, the classical reaction-diffusion equation, much has been accomplished in recent years (see, e.g., [1–9] and the references therein), whereas for degenerated evolutionary equations such information is by comparison very incomplete. The main feature of the problem (1.1) is that the differential operator -div(a(x)∇u) is degenerate because of the presence of a nonnegative diffusion coefficient a(x) which is allowed to vanish somewhere (the physical meaning, see [10–12]). Actually, in order to handle media which have possibly somewhere “perfect” insulators (see [10]) the coefficient a is allowed to have “essential” zeroes at some points or even to be unbounded. In [13], the authors considered the existence of positive solutions when nonlinearity is superlinear and subcritical function for a semilinear degenerate elliptic equation under the assumption that a∈Lloc1(Ω), for some α∈(0,2], satisfies
(1.4)liminfx→z|x-z|-αa(x)>0,foreveryz∈Ω-.
Recently, motivated by [13], under the same assumption as in [13], the authors of [11, 12, 14–20] proved the existence of global attractors of a class of degenerate evolutionary equations for the case of α∈(0,2).The present paper is devoted to the case ofα=2 which is essentially different from the case of α∈(0,2), and which will cause some technical difficulties. In [13], the authors pointed out that the number 2α*=2n/(n-2+α) plays the role of critical exponent. It is well known that some kind of compactness of the semigroup associated with (1.1) is necessary to prove the existence of the global attractor in L2(Ω). However, there is no corresponding compact embedding result in this case since H01,a(Ω) is compactly embedded only into Lr(Ω)(r<2) but not L2(Ω). Hence, the existence of the global attractor in L2(Ω) cannot be obtained by usual methods.In this paper, we assume the weighted functiona satisfies the following. (A1)
a∈L∞(Ω) and liminfx→z|x-z|2a(x)>0 for every z∈Ω-.We will firstly obtain the existence and uniqueness of weak global solutions by use of the singular perturbation then use the asymptotic a priori estimate (see [9]) to verify that the semigroup associated with our problem is asymptotically compact and establish the existence of the global attractor in L2(Ω), Lp(Ω)(p⩾2) and H01,a(Ω), respectively.


## 2. Preliminary Results

In this section, we firstly present some notation and preliminary facts on functional spaces then review some necessary concepts and theorems that will be used to prove compactness of the semigroup. For convenience, hereafter let∥·∥p be the norm of Lp(Ω)(p⩾1), |u| the modular (or the absolute value) of u, and C an arbitrary positive constant, which may vary from line to line and even in the same line.


### 2.1. Functional Spaces

The appropriate Sobolev space for (1.1) is H01,a(Ω), defined as a completion of C0∞(Ω) with respect to the norm
(2.1)∥u∥H01,a=(∫Ω‍a(x)|∇u|2dx)1/2.
The dual space is denoted by H-1,a(Ω), that is, (H01,a(Ω))*=H-1,a(Ω).The next proposition refers to continuous and compact inclusion ofH01,a(Ω).Proposition 2.1 (see [13]).
LetΩ be bounded domain in ℝn (n⩾2) and let a∈Lloc1(Ω) satisfy (1.4) for some α∈(0,2]. Then the following embeddings hold: (i)
H01,a(Ω) is continuously embedded in W01,2n/(n+α)(Ω);(ii)
H01,a(Ω) is continuously embedded in L2α*(Ω);(iii)
H01,a(Ω) is compactly embedded in Lr(Ω) as 1⩽r<2α*=2n/(n-2+α).Remark 2.2.
2
α
*
⩾
2 when α∈(0,2), 2α*=2 when α=2, which plays the role of the critical exponent in the Sobolev embedding.In this paper we only consider the case ofα=2 when n⩾2.


### 2.2. Some Results on Existence of Global Attractors

In this subsection, we review briefly some basic concepts and results on the existence of global attractors; see [2, 5, 7, 9] for more details.Definition 2.3.
Let{S(t)}t⩾0 be a semigroup on Banach space X. {S(t)}t⩾0 is called asymptotically compact if for any bounded sequence {xn}n=1∞ and tn⩾0, tn→∞ as n→∞, and {S(tn)xn}n=1∞ has a convergent subsequence in X.Theorem 2.4.
Suppose{S(t)}t⩾0 is a semigroup on Lp(Ω)(p⩾1). Assume further {S(t)}t⩾0 is a continuous or weak continuous semigroup on Lq(Ω) for some q⩽p and possesses a global attractor in Lq(Ω), where Ω⊂ℝn is bounded. Then {S(t)}t⩾0 possesses a global attractor in Lp(Ω) if and only if (i)
{S(t)}t⩾0 has a bounded absorbing set B0 in Lp(Ω), and(ii)
for anyε>0 and any bounded subset B⊂Lp(Ω), there exist positive constants T=T(ε,B) and M=M(ε,B) such that
(2.2)∫Ω(|S(t)u0|⩾M)‍|S(t)u0|pdx<εforanyu0∈B,t⩾T.Theorem 2.5.
Let{S(t)}t⩾0 be a semigroup on Lp(Ω)(p⩾1) and have a bounded absorbing set in Lp(Ω). Then for any ε>0 and any bounded subset B⊂Lp(Ω), there exist positive constants T=TB and M=M(ε) such that
(2.3)m(Ω(|S(t)u0|⩾M))⩽εforanyt⩾T,u0∈B,
where m(e) (sometimes we also write it as |e|) denotes the Lebesgue measure of e⊂Ω and Ω(|u|⩾M)≜{x∈Ω∣|u(x)|⩾M}.Theorem 2.6.
For anyε>0, the bounded subset B of Lp(Ω)(p⩾1) has a finite ε-net in Lp(Ω) if there exists a positive constant M=M(ε) which depends on ε such that (i)
B has a finite (3M)(q-p)/q(ε/2)p/q-net in Lq(Ω) for some q, q⩾1;(ii)
(2.4)∫Ω(|u|⩾M)‍|u|pdx⩽2-(2p+2)/pεforanyu∈B.


## 2.1. Functional Spaces

The appropriate Sobolev space for (1.1) is H01,a(Ω), defined as a completion of C0∞(Ω) with respect to the norm
(2.1)∥u∥H01,a=(∫Ω‍a(x)|∇u|2dx)1/2.
The dual space is denoted by H-1,a(Ω), that is, (H01,a(Ω))*=H-1,a(Ω).The next proposition refers to continuous and compact inclusion ofH01,a(Ω).Proposition 2.1 (see [13]).
LetΩ be bounded domain in ℝn (n⩾2) and let a∈Lloc1(Ω) satisfy (1.4) for some α∈(0,2]. Then the following embeddings hold: (i)
H01,a(Ω) is continuously embedded in W01,2n/(n+α)(Ω);(ii)
H01,a(Ω) is continuously embedded in L2α*(Ω);(iii)
H01,a(Ω) is compactly embedded in Lr(Ω) as 1⩽r<2α*=2n/(n-2+α).Remark 2.2.
2
α
*
⩾
2 when α∈(0,2), 2α*=2 when α=2, which plays the role of the critical exponent in the Sobolev embedding.In this paper we only consider the case ofα=2 when n⩾2.


## 2.2. Some Results on Existence of Global Attractors

In this subsection, we review briefly some basic concepts and results on the existence of global attractors; see [2, 5, 7, 9] for more details.Definition 2.3.
Let{S(t)}t⩾0 be a semigroup on Banach space X. {S(t)}t⩾0 is called asymptotically compact if for any bounded sequence {xn}n=1∞ and tn⩾0, tn→∞ as n→∞, and {S(tn)xn}n=1∞ has a convergent subsequence in X.Theorem 2.4.
Suppose{S(t)}t⩾0 is a semigroup on Lp(Ω)(p⩾1). Assume further {S(t)}t⩾0 is a continuous or weak continuous semigroup on Lq(Ω) for some q⩽p and possesses a global attractor in Lq(Ω), where Ω⊂ℝn is bounded. Then {S(t)}t⩾0 possesses a global attractor in Lp(Ω) if and only if (i)
{S(t)}t⩾0 has a bounded absorbing set B0 in Lp(Ω), and(ii)
for anyε>0 and any bounded subset B⊂Lp(Ω), there exist positive constants T=T(ε,B) and M=M(ε,B) such that
(2.2)∫Ω(|S(t)u0|⩾M)‍|S(t)u0|pdx<εforanyu0∈B,t⩾T.Theorem 2.5.
Let{S(t)}t⩾0 be a semigroup on Lp(Ω)(p⩾1) and have a bounded absorbing set in Lp(Ω). Then for any ε>0 and any bounded subset B⊂Lp(Ω), there exist positive constants T=TB and M=M(ε) such that
(2.3)m(Ω(|S(t)u0|⩾M))⩽εforanyt⩾T,u0∈B,
where m(e) (sometimes we also write it as |e|) denotes the Lebesgue measure of e⊂Ω and Ω(|u|⩾M)≜{x∈Ω∣|u(x)|⩾M}.Theorem 2.6.
For anyε>0, the bounded subset B of Lp(Ω)(p⩾1) has a finite ε-net in Lp(Ω) if there exists a positive constant M=M(ε) which depends on ε such that (i)
B has a finite (3M)(q-p)/q(ε/2)p/q-net in Lq(Ω) for some q, q⩾1;(ii)
(2.4)∫Ω(|u|⩾M)‍|u|pdx⩽2-(2p+2)/pεforanyu∈B.


## 3. Existence and Uniqueness of the Weak Global Solutions

In this paper, throughout we denoteΩT=Ω×[0,T], V=L2(0,T;H01,a(Ω))∩Lp(ΩT) and V*=L2(0,T;H-1,a(Ω))+Lq(ΩT), respectively, where q is the conjugate exponent of p, that is, 1/p+1/q=1. In addition, we always assume that f satisfies (1.2)-(1.3) and the external forcing term g belongs only to L2(Ω).Definition 3.1.
A functionu(x,t) is called a weak solution of (1.1) on [0,T] if and only if
(3.1)u∈𝒞([0,T];L2(Ω))∩L2(0,T;H01,a(Ω))∩Lp(0,T;Lp(Ω))
and u|t=0=u0 almost everywhere in Ω such that
(3.2)∫ΩT‍(∂u∂tξ+a∇u∇ξ+f(u)ξ)=∫ΩT‍gξ
holds for all test functions ξ∈V.The following lemma makes the initial condition in problem (1.1) meaningful.Lemma 3.2 (see [16]).
Ifu∈V and du/dt∈V*, then u∈𝒞([0,T];L2(Ω)).Theorem 3.3.
AssumeΩ⊂ℝn (n⩾2) is a bounded open domain with smooth boundary, f satisfies (1.2)-(1.3), and g∈L2(Ω). Then for any u0∈L2(Ω) and T>0 there exists a unique solution u of (1.1) which satisfies
(3.3)u∈𝒞([0,T];L2(Ω))∩L2(0,T;H01,a(Ω))∩Lp(ΩT).
The mapping u0↦u(t) is continuous in L2(Ω).Proof.
For any0<ε<1, we choose uε,0∈Cc∞(Ω) such that ∥uε,0∥L∞(Ω) are uniformly bounded with respect to ε, and
(3.4)uε,0→u0inL2(Ω).
Consider the problem(3.5)∂uε∂t-div(aε(x)∇uε)+f(uε)=ginΩ×ℝ+,uε=0on∂Ω×ℝ+,uε(x,0)=uε0inΩ,
where
(3.6)aε(x)=a(x)+ε,x∈Ω.
According to the standard Galerkin methods (see, e.g., [2, 6, 7]), we know the problem (3.5) admits a unique weak solution uε∈𝒞([0,T]; L2(Ω))∩L2(0,T;H01(Ω))∩Lp(0,T;Lp(Ω)). Here uε is called a weak solution of the problem (3.5), if, for any φ∈𝒞0∞(ΩT), we have
(3.7)∫0T‍∫Ω‍(∂uε∂tφ+aε∇uε∇φ+f(uε)φ)dxdt=∫0T‍∫Ω‍gφdxdt
and uε|t=0=uε,0 almost everywhere in Ω.
Now we do some estimates onuε in the following.
Multiplying (3.5) by uε and integrating over Ω, we get
(3.8)12ddt∥uε∥22+∫Ω‍aε(x)|∇uε|2dx+∫Ω‍f(uε)uεdx=∫Ω‍guεdx.
By (1.3) and the Hölder’s inequality, we can deduce that
(3.9)12ddt∥uε∥22+∫Ω‍aε(x)|∇uε|2dx+C1∫Ω‍|uε|pdx⩽C0|Ω|+12C1∥g∥22+C12∥uε∥22,
where |Ω|=∫Ω‍1dx.
Using the Gronwall lemma, for anyT>0, we have the following:
(3.10)uεisuniformlyboundedinL∞(0,T;L2(Ω))withrespecttoε.
Integrating (3.8) and (3.9), both sides between 0 and T, and using the Young’s inequality, we may get by a standard procedure (see, e.g., [2, 6, 7]) that
(3.11)∫ΩT‍a|∇uε|2dxdt⩽∫ΩT‍aε|∇uε|2dxdt=∫ΩT‍a|∇uε|2dxdt+ε∫ΩT‍|∇uε|2dxdt⩽C,∫ΩT‍|uε|pdxdt⩽C,
with C independent of ε.
Noting that (1.3), we obtain
(3.12)∥f(uε)∥Lq(ΩT)q=∫0T‍(∫Ω‍|f(uε)|qdx)dt⩽C∫0T‍(∫Ω‍(1+|uε|p-1)qdx)dt⩽C∫0T‍(∫Ω‍1+|uε|q(p-1)dx)dt.
So we have the following:
(3.13)f(uε)isuniformlyboundedinLq(0,T;Lq(Ω))withrespecttoε.
We now extract a weakly convergent subsequence, denoted also byuε for convenience, with
(3.14)uε⇀uinL2(0,T;H01,a(Ω)),uε⇀uinLp(0,T;Lp(Ω)),f(uε)⇀χinLq(0,T;Lq(Ω)),aε∇uε⇀ϑ→inL2(ΩT,ℝn).
Sincef∈𝒞(ℝ), it follows that
(3.15)f(uε)⇀f(u)inLq(0,T;Lq(Ω)).
Now we show thatu is a weak solution of Problem (1.1). Multiply (3.5) by φ and let ε→0+ to derive
(3.16)∫0T‍∫Ω‍∂u∂tφ+ϑ→·∇φ+f(u)φdxdt=∫0T‍∫Ω‍gφdxdt
for φ∈V.
Therefore, in order to obtain the existence we need only to prove(3.17)∫0T‍∫Ω‍ϑ→·∇φdxdt=∫0T‍∫Ω‍a∇u∇φdxdt,φ∈C0∞(ΩT)
for C0∞(ΩT) is dense in V.
From (3.8) we can obtain
(3.18)∫ΩT‍aε|∇uε|2dxdt=-∫ΩT‍∂uε∂tuεdxdt-∫ΩT‍f(uε)uεdxdt+∫ΩT‍guεdxdt.
Let v∈𝒞([0,T],L2(Ω))∩L2(0,T;H01(Ω)). It is obvious that
(3.19)∫0T‍∫Ω‍aε(∇uε-∇v)·(∇uε-∇v)dxdt⩾0.
Therefore,
(3.20)-∫ΩT‍∂uε∂tuεdxdt-∫ΩT‍aε∇uε∇vdxdt-∫ΩT‍a∇v(∇uε-∇v)dxdt+ε∫ΩT‍∇v(∇uε-∇v)dxdt-∫ΩT‍f(uε)uεdxdt+∫ΩT‍guεdxdt⩾0.
Taking ε→0+ in the above inequality and noticing that
(3.21)ε|∫ΩT‍∇v(∇uε-∇v)dxdt|⩽ε∫ΩT‍|∇v||∇uε|dxdt+ε∫ΩT‍|∇v|2dxdt→0asε→0+,
we arrive at
(3.22)-∫ΩT‍∂u∂tu-∫ΩT‍ϑ→·∇v-∫ΩT‍a∇v(∇u-∇v)-∫ΩT‍f(u)u+∫ΩT‍gu⩾0.
On the other hand, choosing φ=u in (3.7) leads to
(3.23)∫0T‍∫Ω‍ϑ→·∇udxdt=-∫ΩT‍∂u∂tu-∫0T‍∫Ω‍f(u)udxdt+∫0T‍∫Ω‍gudxdt.
Then, it follows from (3.22) and (3.23) that
(3.24)∫0T‍∫Ω‍(ϑ→-a∇v)·(∇u-∇v)dxdt⩾0.
Choosing v=u-λφ with λ>0 in the above inequality, we get
(3.25)∫0T‍∫Ω‍(ϑ→-a∇(u-λφ))·∇φdxdt⩾0,
which implies by letting λ→0+ that
(3.26)∫0T‍∫Ω‍(ϑ→-a∇u)·∇φdxdt⩾0.
If we choose λ<0, we achieve the inequality with opposite sign. Thus
(3.27)∫0T‍∫Ω‍(ϑ→-a∇u)·∇φdxdt=0,
which leads to (3.17). Then u∈𝒞([0,T];L2(Ω)) follows from Lemma 3.2.
Now we will show thatu(0)=u0. Choosing some ϕ∈C1([0,T]; H01,a(Ω)∩Lp(Ω)) with ϕ(T)=0 as a test function and integrating by parts in the t variable we have
(3.28)∫0T‍-〈u,ϕ′〉+〈Au,ϕ〉+〈f(u),ϕ〉ds=∫0T‍〈g,ϕ〉ds+(u(0),ϕ(0)).
Doing the approximations as above yields(3.29)∫0T‍-〈uε,ϕ′〉+(Auε,ϕ)+〈f(uε(s)),ϕ〉ds=∫0T‍〈g,ϕ〉ds+〈uε(0),ϕ(0)〉.
taking limits to conclude that
(3.30)∫0T‍-〈u,ϕ′〉+〈Au,ϕ〉+〈f(u),ϕ〉ds=∫0T‍〈g,ϕ〉ds+(u0,ϕ(0)),
since uε0→u0. Thus u(0)=u0.
Thanks to (1.2), uniqueness and continuous dependence on initial conditions can easily be obtained.We can therefore use these solutions to define a semigroup{S(t)}t⩾0 on L2(Ω) by setting
(3.31)S(t)u0=u(t),
which is continuous on u0 in L2(Ω).


## 4. Existence of Global Attractors

In this section, we prove the existence of the global attractors inL2(Ω), Lp(Ω), and H01,a(Ω), respectively. The following result is the existence of bounded absorbing sets which has been established in [18].Theorem 4.1.
The semigroup{S(t)}t⩾0 possesses bounded absorbing sets in L2(Ω), Lp(Ω), and H01,a(Ω), respectively; that is, for any bounded subset B in L2(Ω), there exists a constant T(∥u0∥2), such that
(4.1)∥u(t)∥22⩽ρ0,∥u(t)∥pp+∫Ω‍a(x)|∇u(t)|2⩽ρ1,
for all t⩾T and u0∈B, where both ρ0 and ρ1 are positive constants independent of B, u(t)=S(t)u0.In order to obtain the existence of a global attractor inL2(Ω) we need to verify that {S(t)}t⩾0 possesses some kind of compactness in L2(Ω), which, however, we cannot obtain by usual methods for lack of the corresponding Sobolev compact embedding results for this case. Here, the new method introduced in [9] is used.LetB0 be the bounded absorbing set in H01,a(Ω), then we can consider our problem only in B0. For H01,a(Ω) is compactly continuous into Lr(Ω) for some 1⩽r<2, we know that B0 is compact in Lr(Ω), and B0 has a finite ε-net in Lr(Ω).Firstly, we give the following useful a priori estimate.Theorem 4.2.
For anyε>0 and bounded subsets B⊂L2(Ω), there exist T=T(ε,B) and M=M(ε) such that
(4.2)∫Ω(|u(t)|⩾M)‍|u(t)|2dx<Cεforanyu0∈B,t⩾T.Proof.
For any fixedε>0, there exists δ>0 such that for any e⊂Ω and m(e)⩽δ we have
(4.3)∫e‍|g(x)|2dx<ε.
Moreover, from Theorem 2.5, we know that there exist T=T(B,ε) and M1=M(ε) such that
(4.4)m(Ω(|u(t)|⩾M1))⩽min{ε,δ}foru0∈B,t⩾T.
In addition, thanks to (1.3), we know f(s)⩾0 when s>(C0/C1)1/p. In the following we assume M=max{M1,(C0/C1)1/p} and t⩾T.
Multiplying (1.1) by (u-M)+ and integrating over Ω, we have
(4.5)12ddt∥(u-M)+∥22+∫Ω‍a(x)|∇(u-M)+|2dx+∫Ω‍f(u)(u-M)+dx=∫Ω‍(u-M)+dx,
where (u-M)+ denotes the positive part of u-M, that is,
(4.6)(u-M)+={u-M,u⩾M,0,u⩽M.
LetΩ1=Ω(u(t)⩾M), then
(4.7)12ddt∥(u-M)+∥22+∫Ω1‍a(x)|∇u|2dx+∫Ω1‍f(u)(u-M)dx=∫Ω1‍g(u-M)dx.
By the Cauchy’s and Hölder’s inequality, we deduce that
(4.8)ddt∥(u-M)+∥22+C∫Ω1‍(u-M)pdx⩽C(∫Ω1‍|g|2dx+∫Ω1‍|u-M|2dx)+C0|Ω1|.
Combining with (4.3)-(4.4) and Lp(Ω)↪L2(Ω)(p⩾2), we get
(4.9)ddt∥(u-M)+∥22+C∫Ω1‍(u-M)2dx⩽Cε.
We apply the Gronwall lemma to infer
(4.10)∥(u-M)+∥22⩽Cε.
Replacing(u-M)+ with (u+M)- and using the same method as above, we obtain
(4.11)∥(u+M)-∥22⩽Cε.
Hence, by (4.10) and (4.11), we have (4.2).According to Theorem2.6, we know B0 is compact in L2(Ω); hence, Theorem 4.1 implies the existence of an attractor in L2(Ω), immediately.Theorem 4.3.
The semigroup{S(t)}t⩾0 associated with (1.1) possesses a global attractor A2 in L2(Ω), that is, A2 is compact and invariant in L2(Ω) and attracts the bounded sets of L2(Ω) in the topology of L2(Ω).We now establish the existence of global attractor inLp(Ω).Theorem 4.4.
The semigroup{S(t)}t⩾0 associated with (1.1) possesses a global attractor Ap in Lp(Ω), that is, Ap is compact and invariant in Lp(Ω) and attracts the bounded sets of L2(Ω) in the topology of Lp(Ω).Proof .
From Theorems2.4, 4.1, and 4.3, we need only to verify that for any ε>0 and bounded subset B⊂L2(Ω) there exist T=T(ε,B) and M=M(ε) such that
(4.12)∫Ω(|u(t)|⩾M)‍|u(t)|pdx<Cεforu0∈B,t⩾T.
LettingF(s)=∫0s‍f(τ)dτ, from (1.3), we deduce that
(4.13)C~1|s|p-k⩽F(s)⩽k+C~2|s|p.
So,
(4.14)C~1∫Ω‍|u|pdx-k|Ω|⩽∫Ω‍F(u)dx⩽k|Ω|+C~2∫Ω‍|u|pdx.
On account of the standard Cauchy’s and Hölder’s inequalities, it follows from (4.7) that
(4.15)ddt∥(u-M)+∥22+C(∫Ω1‍a(x)|∇u|2dx+∫Ω1‍f(u)(u-M)dx)⩽C(∫Ω1‍|u-M|2dx+∫Ω1‍|g|2dx).
Taking t⩾T, integrating the last equality between t and t+1, and combining with (4.2)–(4.4), we have
(4.16)∫tt+1‍(∫Ω1‍a(x)|∇u|2dx+∫Ω1‍f(u)(u-M)dx)ds⩽Cε.
On the other hand, letΩ2=Ω(u⩾2M), multiplying (1.1) by [(u-2M)+]t and integrating over Ω, then we have
(4.17)ddt(∫Ω2‍a(x)|∇u|2dx+∫Ω2‍F(u)dx)⩽Cε.
From (4.16) and (4.17) we apply the uniform Gronwall lemma to obtain
(4.18)∫Ω2‍a(x)|∇u|2dx+∫Ω2‍F(u)dx⩽Cε.
Hence
(4.19)∫Ω2‍F(u)dx⩽Cε.
Replacing(u-M)+ and [(u-2M)+]t with (u+M)- and [(u+2M)-]t, respectively, and repeating the same steps as above we obtain
(4.20)∫Ω(u⩽-2M)‍F(u)dx⩽Cε.
Then, from (4.19)-(4.20), we have
(4.21)∫Ω(|u(t)|⩾2M)‍F(u)dx⩽Cε.
Thus, by (4.4) and (4.14), (4.21) implies (4.12). The proof is finished.


### 4.1. Global Attractor inH01,a(Ω)

In order to prove the existence of a global attractor inH01,a(Ω), we need the following lemma.Lemma 4.5.
For any bounded subsetB in L2(Ω), there exists a constant T=T(B) such that
(4.22)∥ut∥22⩽Mforanyu0∈B,s⩾T,
where ut(s)=(d/dt)(S(t)u0)|t=s, M is independent of B.Proof.
Multiplying (1.1) by ut and integrating over Ω, we get
(4.23)∥ut∥22+12ddt(∫Ω‍a(x)|∇u|2dx+2∫Ω‍F(u)dx)=∫Ω‍gutdx.
Letv=ut and differentiate (1.1) with respect to t to get
(4.24)vt-div(a(x)∇v)+f′(u)v=0.
Multiplying the above equality by v and integrating over Ω, by (1.2), we obtain
(4.25)12ddt∥v∥22+∫Ω‍a(x)|∇v|2dx⩽l∥v∥22.
Taking t⩾T, integrating (4.23) from t to t+1, and considering Theorem 4.1, we get
(4.26)∫tt+1‍∥v∥22dt⩽C(∥g∥22,|Ω|,ρ02)
as t is large enough.
Combing with (4.25) and (4.26) and using the uniform Gronwall lemma, we have
(4.27)∥v(t+1)∥22⩽Cel
as t is large enough.Next, we verify{S(t)}t⩾0 is asymptotically compact in H01,a(Ω).Theorem 4.6.
The semigroup{S(t)}t⩾0 is asymptotically compact in H01,a(Ω).Proof.
LetB0 be an absorbing set in H01,a(Ω) obtained in Theorem 4.1, then we need only to verify that {un(tn)}n=1∞ possesses a convergent subsequence in H01,a(Ω) for any sequence {u0n}n=1∞⊂B0.
In fact, by Theorems4.3 and 4.4, we know {un(tn)}n=1∞ is precompact in L2(Ω) and Lp(Ω). So we can assume that the subsequence {unk(tnk)}k=1∞ is a Cauchy sequence in L2(Ω) and Lp(Ω). Now we prove that {unk(tnk)}k=1∞ is a Cauchy sequence in H01,a(Ω).
Noticing that the prime operatorAau=-div(a(x)∇u) is strong monotone, that is,
(4.28)∥unk(tnk)-unj(tnj)∥H01,a(Ω)2⩽〈Aaunk(tnk)-Aaunj(tnj),unk(tnk)-unj(tnj)〉,
we have
(4.29)∥unk(tnk)-unj(tnj)∥H01,a(Ω)⩽〈-ddtunk(tnk)-f(unk(tnk))+ddtunj(tnj)+f(unj(tnj)),unk(tnk)-unj(tnj)〉⩽∫Ω‍|ddtunk(tnk)-ddtunj(tnj)||unk(tnk)-unj(tnj)|+∫Ω‍|f(unk(tnk))-f(unj(tnj))||unk(tnk)-unj(tnj)|⩽∥ddtunk(tnk)-ddtunj(tnj)∥2∥unk(tnk)-unj(tnj)∥2+C(1+∥unk(tnk)∥pp+∥unj(tnj)∥pp)∥unk(tnk)-unj(tnj)∥p,
which, combining with Lemma 4.5, yields the respected result immediately.In order to establish the existence of global attractor inH01,a(Ω) we need some continuity of the semigroup to guarantee the invariance of global attractor. However, it is difficult to obtain the continuity of semigroup in H01,a(Ω) since we do not impose any restriction on p. Here we use the norm-to-weak continuity instead of the norm-to-norm (or weak-to-weak) continuity of the semigroup in the usual criterions for the existence of global attractors.Theorem 4.7.
The semigroup{S(t)}t⩾0 possesses a global attractor A in H01,a(Ω), that is, A is compact and invariant in H01,a(Ω) and attracts every bounded subset of L2(Ω) in the topology of H01,a(Ω).Proof.
LetB0 be an absorbing set in H01,a(Ω) obtained in Theorem 4.1. Set
(4.30)𝒜=⋂s⩾0‍⋃t⩾s‍S(t)B0¯H01,a(Ω).
Then from Theorems 4.1 and 4.6 we know that 𝒜 is nonempty and compact in H01,a(Ω) and attracts every bounded subset of L2(Ω) in the topology of H01,a(Ω). In addition, it is easy to obtain the norm-to-weak continuity of the semigroup from Theorem 3.2 in [9] which can guarantee that 𝒜 is invariant.
Therefore, the desired claim follows immediately.


## 4.1. Global Attractor inH01,a(Ω)

In order to prove the existence of a global attractor inH01,a(Ω), we need the following lemma.Lemma 4.5.
For any bounded subsetB in L2(Ω), there exists a constant T=T(B) such that
(4.22)∥ut∥22⩽Mforanyu0∈B,s⩾T,
where ut(s)=(d/dt)(S(t)u0)|t=s, M is independent of B.Proof.
Multiplying (1.1) by ut and integrating over Ω, we get
(4.23)∥ut∥22+12ddt(∫Ω‍a(x)|∇u|2dx+2∫Ω‍F(u)dx)=∫Ω‍gutdx.
Letv=ut and differentiate (1.1) with respect to t to get
(4.24)vt-div(a(x)∇v)+f′(u)v=0.
Multiplying the above equality by v and integrating over Ω, by (1.2), we obtain
(4.25)12ddt∥v∥22+∫Ω‍a(x)|∇v|2dx⩽l∥v∥22.
Taking t⩾T, integrating (4.23) from t to t+1, and considering Theorem 4.1, we get
(4.26)∫tt+1‍∥v∥22dt⩽C(∥g∥22,|Ω|,ρ02)
as t is large enough.
Combing with (4.25) and (4.26) and using the uniform Gronwall lemma, we have
(4.27)∥v(t+1)∥22⩽Cel
as t is large enough.Next, we verify{S(t)}t⩾0 is asymptotically compact in H01,a(Ω).Theorem 4.6.
The semigroup{S(t)}t⩾0 is asymptotically compact in H01,a(Ω).Proof.
LetB0 be an absorbing set in H01,a(Ω) obtained in Theorem 4.1, then we need only to verify that {un(tn)}n=1∞ possesses a convergent subsequence in H01,a(Ω) for any sequence {u0n}n=1∞⊂B0.
In fact, by Theorems4.3 and 4.4, we know {un(tn)}n=1∞ is precompact in L2(Ω) and Lp(Ω). So we can assume that the subsequence {unk(tnk)}k=1∞ is a Cauchy sequence in L2(Ω) and Lp(Ω). Now we prove that {unk(tnk)}k=1∞ is a Cauchy sequence in H01,a(Ω).
Noticing that the prime operatorAau=-div(a(x)∇u) is strong monotone, that is,
(4.28)∥unk(tnk)-unj(tnj)∥H01,a(Ω)2⩽〈Aaunk(tnk)-Aaunj(tnj),unk(tnk)-unj(tnj)〉,
we have
(4.29)∥unk(tnk)-unj(tnj)∥H01,a(Ω)⩽〈-ddtunk(tnk)-f(unk(tnk))+ddtunj(tnj)+f(unj(tnj)),unk(tnk)-unj(tnj)〉⩽∫Ω‍|ddtunk(tnk)-ddtunj(tnj)||unk(tnk)-unj(tnj)|+∫Ω‍|f(unk(tnk))-f(unj(tnj))||unk(tnk)-unj(tnj)|⩽∥ddtunk(tnk)-ddtunj(tnj)∥2∥unk(tnk)-unj(tnj)∥2+C(1+∥unk(tnk)∥pp+∥unj(tnj)∥pp)∥unk(tnk)-unj(tnj)∥p,
which, combining with Lemma 4.5, yields the respected result immediately.In order to establish the existence of global attractor inH01,a(Ω) we need some continuity of the semigroup to guarantee the invariance of global attractor. However, it is difficult to obtain the continuity of semigroup in H01,a(Ω) since we do not impose any restriction on p. Here we use the norm-to-weak continuity instead of the norm-to-norm (or weak-to-weak) continuity of the semigroup in the usual criterions for the existence of global attractors.Theorem 4.7.
The semigroup{S(t)}t⩾0 possesses a global attractor A in H01,a(Ω), that is, A is compact and invariant in H01,a(Ω) and attracts every bounded subset of L2(Ω) in the topology of H01,a(Ω).Proof.
LetB0 be an absorbing set in H01,a(Ω) obtained in Theorem 4.1. Set
(4.30)𝒜=⋂s⩾0‍⋃t⩾s‍S(t)B0¯H01,a(Ω).
Then from Theorems 4.1 and 4.6 we know that 𝒜 is nonempty and compact in H01,a(Ω) and attracts every bounded subset of L2(Ω) in the topology of H01,a(Ω). In addition, it is easy to obtain the norm-to-weak continuity of the semigroup from Theorem 3.2 in [9] which can guarantee that 𝒜 is invariant.
Therefore, the desired claim follows immediately.



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*Source: 673605-2012-12-19.xml*