# Global Attractors inH1(ℝN)

Authors: Qiao-zhen Ma; Yong-feng Liu; Fang-hong Zhang
Journal: Discrete Dynamics in Nature and Society
 (2012)

---

## Abstract

We study the existence of global attractors for nonclassical diffusion equations inH1(ℝN). The nonlinearity satisfies the arbitrary order polynomial growth conditions.


---

## Body

## 1. Introduction

In this paper, we investigate the long-time behavior of the solutions for the following nonclassical diffusion equations:(1.1)ut-Δut-Δu+f(x,u)=g(x),x∈ℝN,
with the initial data
(1.2)u(x,0)=u0,x∈ℝN,
where g(x)∈L2(ℝN), and the nonlinearity f(x,u)=f1(u)+a(x)f2(u) satisfies(F1) α1|u|p-β1|u|2≤f1(u)(u)≤γ1|u|p+δ1|u|2,f1(u)u≥0,p≥2, and f1′(u)≥-c,(F2) α2|u|p-β2≤f2(u)(u)≤γ2|u|p+δ2,p≥2, and f2′(u)≥-c,and(A) a∈L1(ℝN)∩L∞(ℝN),a(x)>0,whereαi, βi, γi, δi, i=1,2, and c are all positive constants. Moreover, without loss of generality, we also assume f1(0)=f2(0)=0.In 1980, Aifantis in [1–3] pointed out that the classical reaction-diffusion equation
(1.3)ut-Δu=f(u)+g(x)
does not contain each aspect of the reaction-diffusion problem, and it neglects viscidity, elasticity, and pressure of medium in the process of solid diffusion and so forth. Furthermore, Aifantis found out that the energy constitutional equation revealing the diffusion process is different along with the different property of the diffusion solid. For example, the energy constitutional equation is different, when conductive medium has pressure and viscoelasticity or not. He constructed the mathematical model by some concrete examples, which contains viscidity, elasticity, and pressure of medium, that is the following nonclassical diffusion equation:
(1.4)ut-Δut-Δu=f(u)+g(x).
This equation is a special form of the nonclassical diffusion equation used in fluid mechanics, solid mechanics, and heat conduction theory (see [1–4]). Recently, Aifantis presented a new model about this problem and scrutinized the concrete process of constructing model; the reader can refer to [5] for details.The longtime behavior of (1.1) acting on a bounded domain Ω has been extensively studied by several authors in [6–13] and references therein. In [12] the existence of a global attractor for the autonomous case has been shown provided that the nonlinearity is critical and g(x)∈H-1(Ω). Furthermore, for the non-autonomous, the existence of a uniform attractor and exponential attractors has been scrutinized when the time-dependent forcing term g(x,t) only satisfies the translation bounded domain instead of translation compact, namely, g(x,t)∈Lb2(ℝ,L2(Ω)). A similar problem was discussed in [13] by virtue of the standard method based on the so-called squeezing property. To our best knowledge, the dynamics of (1.1) acting on an unbounded domain ℝN has not been considered by predecessors.As we know, if we want to prove the existence of global attractors, the key point is to obtain the compactness of the semigroup in some sense. For bounded domains, the compactness is obtained by a priori estimates and compactness of Sobolev embeddings. This method does not apply to unbounded domains since the embeddings are no longer compact. To overcome the difficulty of the noncompact embedding, in [14], using the idea of Ball [15], the author proved that the solutions are uniformly small for large space and time variables and then showed that the weak asymptotic compactness is equivalent to the strong asymptotic compactness in certain circumstances. In [16], the authors provided new a priori estimates for the existence of global attractors in unbounded domains and then applied this approach to a nonlinear reaction-diffusion equation with a nonlinearity having a polynomial growth for arbitrary order p-1(p≥2) and with distribution derivatives in homogeneous term. More recently, in [17] the authors achieved the existence of global attractors for reaction-diffusion equations in L2(ℝn), by using the methods presented in [18]. Our purpose in this paper is to study the existence of global attractors of (1.1) on the unbounded domains ℝn, and we borrow the idea of [17, 18]. Our main result is Theorem 4.6.This paper is organized as follows. In Section2, we recall some basic definitions and related theorems that will be used later. In Section 3, we prove the existence of weak solution for nonclassical diffusion equations in H1(ℝN). The main result is stated and proved in Section 4.


## 2. Preliminaries

In this section, we iterate some notations and abstract results.Definition 2.1 (see [18]).
LetM be a metric space, and let A be bounded subsets of M. The Kuratowski measure of noncompactness γ(A) of A defined by
(2.1)γ(A)=inf{δ>0∣Aadmitsafinitecoverbysetswhosediameter≤δ}.Definition 2.2 (see [18]).
LetX be a Banach space, and let {S(t)}t≥0 be a family of operators on X. We say that {S(t)}t≥0 is a continuous semigroup (C0 semigroup) (or norm-to-weak continuous semigroup) on X, if {S(t)}t≥0 satisfies(i)
S(0)=Id (the identity),(ii)
S(t)S(s)=S(t+s),  for all t,s≥0,(iii)
S(tn)xn→S(t)x, if tn→t,xn→x in X (or (iii)S(tn)xn⇀S(t)x, if tn→t,xn→x in X).Definition 2.3 (see [18]).
AC0 semigroup (or norm-to-weak continuous semigroup) {S(t)}t≥0 in a complete metric space M is called ω-limit compact if for every bounded subset B of M and for every ε>0, there is a t(B)>0, such that
(2.2)γ(⋃t≥t(B)‍S(t)B)≤ε.Condition C (see [18]).
For any bounded setB of a Banach space X, there exists a t(B)>0 and a finite dimensional subspace X1 of X such that {∥PmS(t)B∥} is bounded and
(2.3)∥(I-Pm)S(t)x∥<εfort≥t(B),x∈B,
where Pm:X→X1 is a bounded projector.Lemma 2.4 (see [18]).
LetX be a Banach space, and let {S(t)}t≥0 be a C0 semigroup (or norm-to-weak continuous semigroup) in X.(1)
If Condition C holds, the{S(t)}t≥0 is ω-limit compact.(2)
LetX be a uniformly convex Banach space. Then {S(t)}t≥0 is ω-limit compact if and only if Condition C holds.Lemma 2.5 (see [18]).
LetX be a Banach space, and let {S(t)}t≥0 be a C0 semigroup (or norm-to-weak continuous semigroup) in X.(1)
If Condition C holds, the{S(t)}t≥0 is ω-limit compact;(2)
LetX be a uniformly convex Banach space. Then {S(t)}t≥0 is ω-limit compact if and only if Condition C holds.Theorem 2.6 (see [18]).
LetX be a Banach space. Then the C0 semigroup (or norm-to-weak continuous semigroup) {S(t)}t≥0 has a global attractor in X if and only if(1)
there is a bounded absorbing setB⊂X.(2)
{S(t)}t≥0 is ω-limit compact.Lemma 2.7 (see [19]).
LetΦ be an absolutely continuous positive function on ℝ+, which satisfies for some ε>0 the differential inequality
(2.4)ddtΦ(t)+2εΦ(t)≤g(t)Φ(t)+h(t),
for almost every t∈ℝ+, where g and h are functions on ℝ+ such that
(2.5)∫τt‍|g(y)|dy≤m1(1+(t-τ)μ),∀t≥τ≥0,
for some m1≥0 and μ∈[0,1), and
(2.6)supt≥0∫tt+1‍|h(y)|dy≤m2,
for some m2≥0. Then
(2.7)Φ(t)≤βΦ(0)e-εt+ρ,∀t∈R+,
for some β=β(m1,μ)≥1 and
(2.8)ρ=βm2eε1-e-ε.Lemma 2.8 (see [20]).
LetX⊂⊂H⊂Y be Banach spaces, with X reflexive. Suppose that un is a sequence that is uniformly bounded in L2(0,T;X), and dun/dt is uniformly bounded in Lp(0,T;Y), for some p>1. Then there is a subsequence that converges strongly in L2(0,T;H).


## 3. Unique Weak Solution

Theorem 3.1.
Assume(F1), (F2), and (A) are satisfied. Then for any T>0 and u0∈H1(RN), there is a unique solution u of (1.1)-(1.2) such that
(3.1)u∈𝒞1([0,T];H1(ℝN))∩Lp(0,T;Lp(ℝN)).
Moreover, the solution continuously depends on the initial data.Proof.
We decompose our proof into three steps for clarity.
Step  1. For any n∈N, we consider the existence of the weak solution for the following problem in B(0,n)≜Bn⊂RN:
(3.2)ut-Δut-Δu+f(x,u)=g(x),x∈Bn,u(x,0)=u0∈H1(Bn),u|∂Ω=0.
Choose a smooth function χn(x) with
(3.3)χn(x)={1,x∈Bn-1,0,x∉Bn.
SinceBn is a bounded domain, so the existence and uniqueness of solutions can be obtained by the standard Faedo-Galerkin methods; see [6, 8, 11, 16]; we have the unique weak solution
(3.4)un∈𝒞1([0,T];H1(Bn))∩Lp(0,T;Lp(Bn)),un(x,0)=χn(x)u0(x).
Step  2. According to Step 1, we denote (d/dt)un=unt; then un satisfies
(3.5)unt-Δunt-Δun+f(x,un)=g(x),x∈Bn,(3.6)un(x,0)=χn(x)u0(x),(3.7)un|∂Bn=0.
For the mathematical setting of the problem, we denote ∥·∥L2(Bn)≜∥·∥Bn, ∥·∥L1(RN)≜∥·∥1, ∥·∥L2(RN)≜∥·∥, ∥·∥L∞(RN)≜∥·∥∞.
Multiplying (3.5) by un in Bn, using f1(u)u≥0, (F2) and (A), we have
(3.8)12ddt(∥∇un∥Bn2+∥un∥Bn2)+∥∇un∥Bn2≤∫Bn‍a(x)(β2-α2|u|p)dx+∫Bn‍gundx≤β2∥a(x)∥1-∫Bn‍α2a(x)|u|pdx+∥g∥22λ+λ2∥un∥Bn2.
By the Poincaré inequality, for some ν>0, we conclude that
(3.9)12ddt(∥∇un∥Bn2+∥un∥Bn2)+ν(∥∇un∥Bn2+∥un∥Bn2)+∫Bn‍α2a(x)|u|pdx≤β2∥a(x)∥1+∥g∥22λ.
Hence, it follows that
(3.10)∥∇un(T)∥Bn2+∥un(T)∥Bn2+2ν∫0T‍(∥∇un(T)∥Bn2+∥un(T)∥Bn2)+2∫0T‍∫Bn‍α2a(x)|u|pdx≤(2β2∥a(x)∥1+∥g∥2λ)T.
We get the following estimate:
(3.11)supt∈[0,T]∥∇un(t)∥Bn2+∥un(t)∥Bn2≤C,∫0T‍(∥∇un(t)∥Bn2+∥un(t)∥Bn2)≤C,∫0T‍∫Bn‍α2a(x)|u(t)|pdx≤C.
Similar to (3.9), using (F1), (F2), and (A), we get
(3.12)∫0T‍∫Bn‍|u(t)|pdx≤C,
where C is independent of n.
(
F
1
) and (F2) yield
(3.13)|f1(un)|≤C(|un|p-1+|un|),|f2(un)|≤C(|un|p-1+1).
Choose q such that (1/p)+(1/q)=1; then (p-1)q=p. Noting that p≥2, then 1<q≤2, and we have the embedding Lp(Bn)↪Lq(Bn). According to (3.12) and (3.13), we get
(3.14)∫0T‍∫Bn‍|f1(u)|q≤C∫0T‍∫Bn‍(|un|p-1+|un|)qdxdt≤C∫0T‍∫Bn‍|un|(p-1)qdxdt+C∫0T‍∫Bn‍|un|qdxdt≤C∫0T‍∫Bn‍|un|p+C∫0T‍∫Bn‍|un|pdxdt≤C,∫0T‍∫Bn‍|f2(u)|q≤C∫0T‍∫Bn‍|a(x)|q(|un|p-1+1)qdxdt≤C|a(x)|∞q-1∫0T‍∫Bn‍a(x)(|un|(p-1)q+1)dxdt≤C|a(x)|∞q-1(C|a(x)|1+∫0T‍∫Bn‍a(x)|un|pdxdt)≤C,
where C is independent of n.
Thanks to (3.14), f1(un) is bounded in Lp(0,T;Lq(Bn)), and af2(un) is bounded in Lp(0,T;Lq(Bn)).
For∀v∈L2(0,T;H01(Bn)),
(3.15)∫0T‍∫Bn‍-Δunv=∫0T‍∫Bn‍∇un∇v≤(∫0T‍∥∇un∥Bn2)1/2(∫0T‍∥∇v∥Bn2)1/2≤(∫0T‍∥∇un∥2)1/2(∫0T‍∥∇v∥Bn2)1/2≤C∥∇v∥H01(Bn),
where C is independent of n. We can obtain that -Δun is bounded in L2(0,T;H-1(Bn)).
Sinceg(x)∈L2(ℝN),
(3.16)g(x)∈L2(0,T;ℝN).
Therefore, there exists s>0, such that L2(0,T;H-1(Bn)), L2(0,T;H01(Bn)), Lq(0,T;Lq(Bn)), and L2(0,T;L2(Bn)) are continuous embedding to Lq(0,T;H-s(Bn)).
According to (3.5) and (3.14)–(3.16), we obtain
(3.17)unt-Δunt∈Lq(0,T;H-s(Bn)).
So un has a subsequent (we also denote un) weak* convergence to u in L2(0,T;H-1(Bn)) and L2(0,T;L2(Bn)); unt-Δunt has a subsequent (we also denote unt-Δunt) weak* convergence to ut-Δut. Let un=0 outside of Bn; we can extend un to ℝN.
As introduced in [6, 20], Cc∞(ℝN) is dense in the dual space of H-1(Bn), L2(Bn), Lq(Bn), and H-s(Bn), so we can choose for all ϕ∈L2(0,T;Cc∞(ℝN))∩Lq(0,T;Cc∞(ℝN)) as a test function such that
(3.18)〈Δun,ϕ〉→〈Δu,ϕ〉,〈unt-Δunt,ϕ〉→〈ut-Δut,ϕ〉.
Since for allϕ∈Cc∞(ℝN), there exists bounded domain Ω⊂ℝN such that ϕ=0, x∉Ω. It follows that un is uniformly bounded in L2(0,T;H01(Ω)), and unt-Δunt∈Lq(0,T;H-s(Ω)). Since H01(Ω)⊂⊂L2(Ω)⊂H-s(Ω), according to Lemma 2.8, there is a subsequence un (we also denote un) that converges strongly to u in L2(0,T;L2(Ω)).
Using the continuity off1 and f2, we have
(3.19)〈f1(un),ϕ〉→〈f1(u),ϕ〉,〈a(x)f2(un),ϕ〉→〈a(x)f2(u),ϕ〉.
In line with (3.18) and (3.19), and let n→∞, we geting for all ϕ∈L2(0,T;Cc∞(ℝN))∩Lq(0,T;Cc∞(ℝN)):
(3.20)〈ut-Δut-Δu+f1(u)+a(x)f2(u),ϕ〉=〈g(x),ϕ〉.
Thus, u is the weak solution of (3.2) and satisfies
(3.21)u∈𝒞1([0,T];H1(ℝN))∩Lp(0,T;Lp(ℝN)).
Step  3 (uniqueness and continuous dependence). Let u0, v0 be in H1(ℝN), and setting w(t)=u(t)-v(t), we see that w(t) satisfies
(3.22)wt-Δwt-Δw+f1(u)-f1(v)+a(x)(f2(u)-f2(v))=0,x∈ℝN.
Taking the inner product with w of (3.22), using (F1), (F2), and (A), we obtain
(3.23)12ddt(∥∇w∥2+∥w∥2)+∥∇w∥2≤|∫‍(f1(u)-f1(v))wdx|+|∫‍a(x)(f2(u)-f2(v))wdx|≤C(1+∥a∥∞)∥w∥2.
By the Gronwall Lemma, we get
(3.24)∥∇w(t)∥2+∥w(t)∥2≤eCt(∥∇w(0)∥2+∥w(0)∥2).
This is uniqueness and is continuous dependence on initial conditions.
Thanks to Theorem3.1, and leting S(t)u0=u(t), S(t):H1(ℝN)→H1(ℝN) is a C0 semigroup.


## 4. Global Attractor inℝN

Lemma 4.1.
Assume(F1), (F2), and (A) are satisfied. There is a positive constant ρ1 such that for any bounded subset B⊂H1(ℝN), there exists T1=T1(B) such that
(4.1)∥∇u(t)∥≤ρ1,∀t≥T1,u0∈B.Proof.
Multiplying (1.1) by u in ℝN, using f1(u)u≥0, (F2) and (A), we have
(4.2)12ddt(∥∇u∥2+∥u∥2)+∥∇u∥2≤∫ℝN‍a(x)(β2-α2|u|p)dx+∫ℝN‍gudx≤β2∥a(x)∥1-∫ℝN‍α2a(x)|u|pdx+∥g∥22λ+λ2∥u∥Bn2.
By virtue of the Poincaré inequality, for some ν>0, there holds
(4.3)12ddt(∥∇u∥2+∥u∥2)+ν(∥∇u∥2+∥u∥2)+∫ℝN‍α2a(x)|u|pdx≤β2∥a(x)∥1+∥g∥22λ.
Furthermore,
(4.4)12ddt(∥∇u∥2+∥u∥2)+ν(∥∇u∥2+∥u∥2)≤β2∥a(x)∥1+∥g∥22λ.
By the Gronwall Lemma, we get
(4.5)∥∇u(t)∥2+∥u(t)∥2≤e-νt(∥∇u(0)∥2+∥u(0)∥2)+2β2∥a(x)∥1+∥g∥2λ.
We completed the proof.According to Lemma4.1, we know that
(4.6)ℬ0={u∈H1(ℝN):∥∇u∥≤ρ}
is a compact absorbing set of a semigroup of operators {S(t)}t≥0 generalized by (1.1)-(1.2), (F1), (F2), and (A).Lemma 4.2.
Assume(F1), (F2), and (A) hold. Then for any u0∈H1(ℝN) and ɛ>0, there are some T(ɛ) and k(ɛ) such that
(4.7)∫|x|≥2k‍|∇u(t)|2dt≤Cɛ,
whenever k≥T(ɛ) and t≥t(ɛ).Proof.
Choose a smooth functionθ(x) with
(4.8)θ(x)={0,0≤s≤1,1,s≥2,
where 0≤θ(s)≤1,  1≤s≤2, and there is a constant c such that |θ′(s)|≤c.
Multiplying (1.1) with θ2(|x|2/k2)u and integrating on ℝN, we obtain
(4.9)12ddt∫ℝN‍θ2(|x|2k2)(|∇u|2+|u|2)dx-∫ℝN‍θ2(|x|2k2)uΔudx=-∫ℝN‍θ2(|x|2k2)f1(u)udx-∫ℝN‍θ2(|x|2k2)a(x)f2(u)udx+∫ℝN‍θ2(|x|2k2)ugdx≤-∫ℝN‍θ2(|x|2k2)f1(u)udx-∫ℝN‍θ2(|x|2k2)a(x)f2(u)udx+λ2∫ℝN‍θ2(|x|2k2)|u|2dx+12λ∫ℝN‍|g|2dx.
Next we deal with the right hand side of (4.9) one by one:
(4.10)∫ℝN‍θ2(|x|2k2)uΔudx=-∫ℝN‍θ2(|x|2k2)|∇u|2dx-∫ℝN‍4xk2θ(|x|2k2)θ′(|x|2k2)u∇udx.
According to the condition |θ′(s)|≤c and the bounded absorbing set in H1(ℝN) for t≥t*, it follows that
(4.11)|∫ℝN‍4xk2θ(|x|2k2)θ′(|x|2k2)u∇udx|=|∫k≤|x|≤2k‍4xk2θ(|x|2k2)θ′(|x|2k2)u∇udx|≤42k∫k≤|x|≤2k‍θ2(|x|2k2)|u||∇u|dx≤22k(∫ℝN‍θ2(|x|2k2)|∇u|2dx+∫ℝN‍|u|2dx)≤Ck∫ℝN‍θ2(|x|2k2)|∇u|2dx+Ck,
where C is independent of k. For any 0<ɛ<1 given, let
(4.12)k1(ɛ)=Cɛ.
Hence, combining (4.10) with (4.11), when k≥k1(ɛ), we conclude that
(4.13)∫ℝN‍θ2(|x|2k2)uΔudx≤-12∫ℝN‍θ2(|x|2k2)|∇u|2dx+ɛ.
Using f1(u)u≥0 and (F2), it yields
(4.14)-∫ℝN‍θ2(|x|2k2)f1(u)udx-∫ℝN‍θ2(|x|2k2)a(x)f2(u)udx≤∫ℝN‍θ2(|x|2k2)a(x)(β2-α2|u|p)dx≤β2∫ℝN‍θ2(|x|2k2)a(x)dx≤β2∫|x|≥k‍a(x)dx.
Since a∈L1(ℝN), there exist k2(ɛ)>k1(ɛ), such that
(4.15)∫|x|≥k‍a(x)dx≤ɛ2β2.
Then
(4.16)-∫ℝN‍θ2(|x|2k2)f1(u)udx-∫ℝN‍θ2(|x|2k2)a(x)f2(u)udx≤ɛ2.
From the assumption g(x)∈L2(ℝN), provide k≥k(ɛ)≥k2(ɛ), such that
(4.17)∫|x|≥k‍|g|2dx≤ɛλ.
Thus combining (4.9), (4.13), (4.16), and (4.17), we finally obtain
(4.18)ddt∫ℝN‍θ2(|x|2k2)(|∇u|2+|u|2)dx+∫ℝN‍θ2(|x|2k2)|∇u|2dx≤4ɛ.
Furthermore, there holds
(4.19)ddt∫ℝN‍θ2(|x|2k2)(|∇u|2+|u|2)dx+∫ℝN‍θ2(|x|2k2)(|∇u|2+|u|2)dx≤2∫ℝN‍θ2(|x|2k2)(|∇u|2+|u|2)dx+4ɛ.
According to Lemma 2.7, we obtain
(4.20)∫ℝN‍θ2(|x|2k2)(|∇u(t)|2+|u(t)|2)≤β∫ℝN‍θ2(|x|2k2)(|∇u(0)|2+|u(0)|2)e-t/2+βe1/21-e-1/2ɛ.
Thus, we get
(4.21)∫|x|≥2k‍|∇u(t)|2dt≤∫ℝN‍θ2(|x|2k2)(|∇u(t)|2+|u(t)|2)≤Cɛ,
provided T≥T(ɛ) and k≥k~(ɛ), we complete the proof.Lemma 4.3.
Assume(F1), (F2), and (A) hold. There is a positive constant ρ2 such that for any bounded subset B⊂H2(ℝN), there exists T2=T2(B) such that
(4.22)∥Δu(t)∥≤ρ2,∀t≥T2,u0∈B.Proof.
Multiplying (1.1) by -Δu in ℝN, we find
(4.23)12ddt(∥∇u∥2+∥Δu∥2)+∥Δu∥2=∫ℝN‍f1(u)Δudx+∫ℝN‍a(x)f2(u)Δudx-∫ℝN‍gΔudx.
Using (F1), (F2), and (A), we have the following estimates:
(4.24)∫ℝN‍f1(u)Δudx≤∫ℝN‍f1′(u)|∇u|2dx≤c∥∇u∥2,∫ℝN‍a(x)f2(u)Δudx≤∫ℝN‍a(x)f2′(u)|∇u|2dx≤c∥∇u∥2,|∫ℝN‍gΔudx|≤c∥g(x)∥2+12∥Δu∥2.
Together with (4.6) and (4.19)–(4.21), by the Poincaré inequality, for some μ>0, this yields
(4.25)ddt(∥∇u∥2+∥Δu∥2)+μ(∥∇u∥2+∥Δu∥2)≤C∥g(x)∥2+C.
By the Gronwall Lemma, we get
(4.26)∥∇u(t)∥2+∥Δu(t)∥2≤e-μt(∥∇u(0)∥2+∥Δu(0)∥2)+C.
We complete the proof.Remark 4.4.
There is a constantC>0, such that for any bounded subset B⊂B(0,ρ2)⊂H1(ℝN), when t>t*, there holds
(4.27)∫tt+1‍(∥∇u∥2+∥Δu∥2)≤C.Lemma 4.5.
Assume(F1), (F2), and (A) are satisfied. Then the semigroup {S(t)}t≥0 associated with the initial value problems (1.1) and (1.2) is ω-limit compact.Proof.
DenoteBR=B(0;R)∩ℝN, and we split u(t) as
(4.28)u(t)=χ(x)u(t)+(1-χ(x))u(t)=u1(t)+u2(t),
where θ(x) is a smooth function:
(4.29)χ(x)={1,x∈BR,0,x∉BR+1,
with 0≤χ(x)≤1, and there is a positive constant c such that |χ′(x)|≤c. Then
(4.30)u1(t)={u(t),x∈BR,0,x∉BR+1,χ(x)u(t),others,u2(t)={0,x∈BR,u(t),x∉BR+1,(1-χ(x))u(t),others.
From Lemma 4.1, we know that u1(t)∈H1(BR) as t≥T1.
For anyɛ>0 given, we can choose R large enough; by Remark 4.4, we can assume
(4.31)∫|x|≥R‍|∇u|2dx≤ɛ2.
So we conclude that
(4.32)∥∇u2∥2≤ɛ2.
For any bounded set B⊂H1(ℝN), {S(t)B}t≥0={S(t)u0∣u0∈B}t≥0 can be split as
(4.33)S(t)B=χ(x)s(t)B+(1-χ(x))s(t)B.
Then in line with the property of noncompact measure, it follows that
(4.34)γ(S(t)B)=γ(χ(x)s(t)B)+γ((1-χ(x))s(t)B).
On the other hand,
(4.35)γ(χ(x)s(t)B)={χ(x)s(t)u0=u1(t)∣u0∈B}.
From Lemma 4.3, we get
(4.36)∥u1∥H2(BR+1)≤C,∀t>t*+1.
Recall that
(4.37)(1-χ(x))s(t)B={(1-χ(x))s(t)u0=u2∣u2∈B}.
On account of Remark 4.4, it yields
(4.38)γ((1-χ(x))s(t)B)≤ɛ,∀t>t*+1.
Therefore, we have
(4.39)γ(S(t)B)B≤ɛ,∀t>t*+1.
That is, {S(t)}t≥0 is ω-limit compact in H1(ℝN).Theorem 4.6.
Assume(F1), (F2), and (A) hold. Then the semigroup {S(t)}t≥0 associated with the initial value problems (1.1) and (1.2) has a global attractor 𝒜 in H1(ℝN).



---

*Source: 672762-2012-11-29.xml*