# Global Attractor for a Chemotaxis Model with Reaction Term

Authors: Xueyong Chen; Jianwei Shen
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

A chemotaxis model with reproduction term in a bounded domainΩ⊂Rn is discussed in this paper, where n=2,3. First, the existence of a global-in-time solution is given, and then a global attractor for this model is obtained.


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## Body

## 1. Introduction

This paper deals with global attractor of a quasilinear parabolic system introduced in [1] by Horstmann and Winkler to model chemotaxis. Chemotaxis phenomenon is quite common in biosystem. The survival of many organisms, from microscopic bacteria to the largest mammals, depends on their ability to navigate in a complex environment through the detection, integration, and processing of a variety of internal and external signals. This movement is crucial for many aspects of behavior, including the location of food sources, avoidance of predators, and attracting mates. The ability to migrate in response to external signals is shared by many cell populations. The directed movement of cells and organisms in response to chemical gradients is called chemotaxis.The classical chemotaxis model has been extensively studied in the last few years (see [1–5]); Payne and Straughan [4] tackle precise nonlinear decay for classical model. Decay for a nonlinear chemotaxis system modeling glia cell movement in the brain is treated by Quinlan and Straughan [5].In this paper, we are concerned with the following chemotaxis model:(1)∂u∂t=▵u-∇·((u+1)α∇g(w))+f(u,w),x∈Ω,t>0,∂w∂t=▵w+u-w,x∈Ω,t>0,∂u∂n=∂w∂n=0,x∈∂Ω,u(x,0)=u0(x),w(x,0)=w0(x),x∈Ω-,
where u(x,t) represents the density of a biological species, which could be a cell, a germ, or an insect, while w(x,t) denotes the concentration of a chemical substance on position x∈Ω and at time t∈[0,∞). α is a positive constant. Ω is an open bounded subset of Rn with the boundary of class C3. g(s), f(s) are two real smooth functions for s∈[0,+∞) satisfying the following conditions: (i)
f(0,w)=0;(ii)
there existsM>0 such that f(u,w)≤0 as u>M;(iii)
there existsλ>0 such that |g′(s)|<λ for all s∈[0,+∞).g′(w)(u+1)α denotes a chemotactic sensitivity function. (u+1)α can be replaced by (u+ε)α, where ε is an arbitrary positive constant. For convenience, we choose ε=1. f(u,w) describes the intrinsic rate of growth of cells population. For α=1,ε=0,andg(w)=w, system (1) equals the most common formulation of the Keller-Segel model.To our knowledge it has never been analyzed whether the global attractor of system (1) exists if the positive power α≠1; this system is more motivated from the mathematical point of view than from the biological one, but it will help to get more insights in the understanding of the behavior of the problem. We will look at this aspect of the critical exponent α which decides whether global attractor can exist or not.The existence of the global attractor for semilinear reaction diffusion equations in bounded and unbounded domains has been studied extensively [6, 7]. However, chemotaxis model is always a strongly coupled quasilinear parabolic system. There are a few articles to discuss the global attractor of such a system. We construct a local solution to (1) by the semigroup method and then discuss its regularity by a priori estimate method we set up for a strongly coupled quasilinear parabolic system.For readers' convenience, the following standard result on attractor is first presented here (see [3, 6–9]).Proposition 1.
Suppose thatX is a metric space and S(t)t≥0 is a semigroup of continuous operators in X. If S(t)t≥0 has a bounded absorbing set and is asymptotically compact, then S(t)t≥0 possesses a global attractor which is a compact invariant set and attracts every bounded set in X.Definition 2.
The semigroupS(t) is asymptotically compact; that is, if un is bounded in X and tn→∞, then S(tn)un is precompact in X.
We first show that there is a unique global solution to (1) for any nonnegative initial functions u0(x)∈L2(Ω), w0(x)∈W1,2(Ω). And then we define a continuous semigroup S(t)t≥0 in L2(Ω)×W1,2(Ω). Next, we find a bounded absorbing set and prove that the semigroup S(t) is asymptotically compact. Thus, from the standard result, it is shown that there is a global attractor to (1) in L2(Ω)×W1,2(Ω).


## 2. Preliminary

Some well-known inequalities and embedding results that will be used in the sequel are presented.Lemma 3 (see [10]).
Ifp,q≥1 and p(n-q)<nq, then, for r∈(0,p),
(2)∥u∥Lp(Ω)≤c∥u∥W1,qa·∥u∥Lr(Ω)(1-a)∀u∈W1,q(Ω),
where a=(n/r-n/p)/(1-n/q+n/r)∈(0,1).Lemma 4 (see [10]).
Let1≤q≤p≤∞ and f∈Lq(Ω). Then
(3)∥et∆f∥p≤(4πt)(-n/2)(1/q-1/p)∥f∥q,∥∇et∆f∥p≤ct(-n/2)(1/q-1/p)-1/2∥f∥q,∥(-∆+1)βet∆f∥p≤ct-β(-n/2)(1/q-1/p)e(1-μ)t∥f∥qp≠q,
where β>0, μ>0, and c is a positive constant depending only on p,q,Ω.
(
e
t
▵
f
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(
x
)
=
∫
Ω
‍
G
(
x
-
y
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f
(
y
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d
y and G(x,t) is the Green function of the heat equation ∂u/∂t-▵u=0,x∈Ω,t>0 with the homogeneous Neumann boundary condition ∂u/∂n=0.Lemma 5 (see [11]).
LetAp=-▵ and D(Ap)={φ∈W(Ω)2,p|∂φ/∂N|∂Ω=0}. Then
(4)D((Ap+1)β)↪W1,p(Ω),ifβ>12,D((Ap+1)β)↪Cδ(Ω),if2β-np>δ≥0,∥(A+1)βe-t(A+1)u∥Lp(Ω)≤ct-β∥u∥Lp(Ω).Lemma 6.
Letr and s be nonnegative real numbers satisfying r+s<2. Then for any ε>0 there exists a constant cε>0 such that
(5)arbs≤ε(a2+b2)+cε∀a,b>0.Lemma 7 (see [2]).
Suppose thatHs is an interpolation space of Hs0 and Hs1, where 0≤s0<s1<∞, 0<θ<1, s=(1-θ)s0+θs1, and then
(6)∥·∥Hs≤c∥·∥Hs01-θ∥·∥Hs1θ.
Here Hs(Ω), s>0, denotes the fractional Sobolev space in Ω.Lemma 8 (see [1]).
Letβ>0, p∈(1,+∞), for all ε>0; there exists c(ε)>0 for any w∈Lp(Ω), such that
(7)∥(A+1)βe-tA∇·w∥L(Ω)p≤c(ε)t-β-(1/2)-ε∥w∥L(Ω)p.


## 3. Local Existence and Uniqueness

The local existence of a solution to system (1) is discussed in this section. First, an estimate to w in (1) is shown.Lemma 9.
Assume that there exist1≤q≤p≤∞, (1/q)-(1/p)<1/n, and u∈L∞([0,∞);Lq(Ω)). Then for any τ>0(8)∥w(t)∥W1,p≤τ-α∥w0∥L1+cΓ(γ)supτ<s<t∥u(s)∥q∀t≥τ,
where α>0, γ=1-β-(n/2)((1/q)-(1/p)), Γ(·) is the Gamma function, and
(9)w(x,t)=et(▵-1)w0+∫0t‍e(t-s)(▵-1)u(s)ds.
The proof of the lemma can be found in another paper [3].Theorem 10.
Suppose0≤u0(x)∈L2(Ω), 0≤w0(x)∈W1,2(Ω), and α<1/n. Then there is a T≤∞ (depending on ∥u0∥L2(Ω),∥w0∥W1,2(Ω)) such that there exists a unique nonnegative solution (u(x,t),w(x,t)) to (1) in [0,T] and
(10)u∈C0([0,T];L2(Ω))⋂C2,1(Ω;(0,T]),w∈C0([0,T];W1,2(Ω))⋂C2,1(Ω;(0,T])⋂C0((0,T];C3(Ω)).Proof.
ChooseT∈(0,1) and R>0 to be fixed. In Banach space X=C0([0,T];L2(Ω))×C0([0,T];W1,2(Ω)), we define a bounded closed set
(11)S:={(u,w)∈X∣∥u,w∥X≤R}.
Let
(12)ψ(u,w)(t)=(ψ1(u,w)(t)ψ2(u,w)(t))=(et▵u0-∫0t‍e(t-s)▵[∇(uα∇g(w))+f(u)]dset(▵-1)w0+∫0t‍e(t-s)(▵-1)u(s)ds),
where f(u)=u(a-bu).
Next, we prove thatψ is a contractive mapping from S into itself for T small enough and R sufficiently large. By Lemma 4, then
(13)∥ψ1(u,w)∥L2≤∥e-tAu0(x)∥L2+c∫0t‍∥e-(t-s)A∇(uα∇g(w))∥L2ds+∫0t‍∥e-(t-s)Af(u(s))∥L2ds≤∥u0(x)∥L2+c∫0t‍(t-s)(-n/2)((1/q)-(1/2))-(1/2)∥uα∇g(w)∥Lqds+∫0t‍(t-s)(-n/2)(1-(1/2))∥f(u(s))∥L1ds,
where A=-▵. Since α<1/n, there is a q∈[2n/(2+n),2), such that 2qα/(2-q)≤2 and 1/2-(n/2)((1/q)-(1/2))>0. By Hölder’s inequality
(14)∥uα∇g(w)∥Lq≤λ∥uα∇w∥Lq≤c(λ)∥u∥L2qα/(2-q)α∥∇w∥L2≤c(λ)∥u∥L2α∥∇w∥L2≤c(λ)R2,
then
(15)c∫0t‍(t-s)(-n/2)((1/q)-(1/2))-(1/2)∥uα∇g(w)∥Lqds≤c(λ)R2T(1/2)-(n/2)((1/q)-(1/2)).
Similarly, forn≤3, there is ∫0t(t-s)(-n/2)(1-(1/2))∥f(u(s))∥L1ds≤RT1-(n/4). Hence
(16)∥ψ1(u,w)∥L2≤∥u0(x)∥L2+cR2T1/2-(n/2)((1/q)-(1/2))+RT1-(n/4).
By Lemma5, for any t∈[0,T), there is a γ∈(1/2,1) such that
(17)∥ψ2(u,w)∥W1,2≤∥e-t(A+1)w0(x)∥W1,2+∫0t‍∥e-(t-s)(A+1)u(s)∥W1,2ds≤∥w0(x)∥W1,2+c∫0t‍∥(A+1)γe-(t-s)(A+1)u(s)∥L2ds≤∥w0(x)∥W1,2+c∫0t‍(t-s)-γ∥u(s)∥L2ds≤∥w0(x)∥W1,2+cRT1-γ.
Equations (16) and (17) imply that ψS⊂S for any fixed positive R large enough and T small enough. Now we show that ψ is a contractive operator from S to S. For for all (u,w),(u-,w-)∈S,
(18)∥ψ1(u,w)-ψ1(u-,w-)∥L2≤c∫0t‍∥e-(t-s)A[∇(uα∇g(w))-∇(u-α∇g(w-))]∥L2ds+∫0t‍∥e-(t-s)A(f(u)-f(u-))∥L2ds≤c∫0t‍(t-s)(-n/2)((1/q)-(1/2))-(1/2)∥uα∇w-u-α∇w-∥Lqds+∫0t‍(t-s)-n/4∥f(u)-f(u-)∥L1ds≤cR2(T1/2-(n/2)((1/q)-(1/2))+T1-n/4)∥(u,w)-(u-,w-)∥X∀t∈[0,T),∥ψ2(u,w)-ψ2(u-,w-)∥W1,2≤c∫0t‍∥(A+1)γe-(t-s)(A+1)(u(s)-u-(s))∥L2ds≤c∫0t‍(t-s)-γ∥u(s)-u-(s)∥L2ds≤cT1-γ∥(u,w)-(u-,w-)∥X∀t∈[0,T).
Equation (18) implies that ψ is a contractive mapping if T is sufficiently small. By Banach's fixed point theorem, there exists a unique fixed point (u,w)∈X which is just a local solution to (1) in X.
Sinceu0(x)≥0, w0(x)≥0, then, for any given smooth function w, u=0 is the subsolution of the following problem:
(19)∂u∂t-▵u+∇·((u+1)α∇g(w))-u(a-bu)=0,x∈Ω,t>0,∂u∂n=0,x∈∂Ω,u(x,0)=u0(x),x∈Ω-
and, for any given smooth function u, w=0 is the subsolution of the following problem:
(20)∂w∂t-▵w+w=u,x∈Ω,t>0,∂w∂n=0,x∈∂Ω,w(x,0)=w0(x),x∈Ω-.
By the comparison principle, for any t∈[0,Tmax) and x∈Ω¯, u(x,t)≥0; w(x,t)≥0.
Now, we discuss the regularity of the solution to (1). From the previous analysis, u(t,x) is bounded in L2(Ω) for any t∈[0,T]. Then by Lemma 9, for any τ∈(0,T], we deduce that w(t)∈W1,p(Ω), for all t∈[τ/2,T], where n<p<2n/(n-2) and n=2,3. From the semigroup representation of the solution to (1) and Lemmas 3 and 4, for any q∈(2,n/(n-2)), t∈[τ,T],
(21)∥u(x,t)∥Lq≤∥e-(t-τ/2)Auτ/2(x)∥Lq+c∫τ/2t‍∥e-(t-s)A∇((u+1)α∇g(w))∥Lqds+∫τ/2t‍∥e-(t-s)Af(u(s))∥Lqds≤(t-τ2)(-n/2)((1/2)-(1/q))∥uτ/2(x)∥L2+cλ∫τ/2t‍(t-s)(-n/2)((1/2)-(1/q))∥(u+1)α∇w∥L2ds+∫τ/2t‍(t-s)(-n/2)(1-(1/q))∥f(u(s))∥L1ds.
Since α<1/n, then there exists p1∈(n/2,1/α] such that 2αp1≤2, 2p1′<2n/(n-2) and
(22)∥(u+1)α∇w∥L2≤c∥u∥L2αp1α∥∇w∥L2p1′≤c∥u∥L2α∥∇w∥L2p1′.
Here 1/p1+1/p1′=1. Then
(23)∥u(x,t)∥Lq≤cτ(-n/2)((1/2)-(1/q))∥u0(x)∥L2+csupτ/2<s<t∥u∥L2∥∇w∥L2p1′∫τ/2t‍(t-s)(-n/2)((1/2)-(1/q))-(1/2)ds+c∫τ/2t‍(t-s)(-n/2)(1-1/q)dst∈[τ,T]
which implies that, for any q∈(2,n/(n-2)), ∥u(x,t)∥Lq is bounded for all t∈[τ,T]. By using Lemma 9 again and repeating the above process, it can be proved that
(24)u∈C0((τ,T];L∞(Ω)),w∈C0((τ,T];C0(Ω)).
There exist p, β, and ε satisfying p>n, 1/2>β>n/2p, and ε<1/2-β such that, for any small enough constant η>0,
(25)∥u(x,t)∥C0≤∥e-(t-τ)Auτ(x)∥C0+c∫τt‍∥e-(t-s)A∇((u+1)α∇g(w))∥C0ds+∫τt‍∥e-(t-s)Af(u(s))∥C0ds≤∥(A+1)βe-(t-τ)Auτ(x)∥Lp+cλ∫τt‍∥(A+1)βe-(t-s)A∇((u+1)α∇w)∥Lpds+∫τt‍∥(A+1)βe-(t-s)Af(u(s))∥Lpds≤η(-n/2)((1/2)-(1/p))e(1-μ)t∥uτ(x)∥L2+csupτ<s<t∥(u+1)α∇w∥Lp∫τt‍(t-s)-β-1/2-εds+c∫τt‍(t-s)-β-ε1dst∈[τ+η,T),
where ε1=(n/2)((1/q)-(1/p))≤1-β, which implies that u∈C0((0,T];C0(Ω)). In conclusion, we see that
(26)u(x,t)∈C([0,T];L2(Ω))⋂C2,1(Ω;(0,T]),w(x,t)∈C([0,T];W1,2(Ω))⋂C2,1(Ω;(0,T])⋂C((0,T];C3(Ω)).
By semigroup techniques and Schauder estimates (Theorem IV. 5.1–5.3 in [7] and Lemmas 3.2–3.3 in [10]), we have that
(27)u∈C0([0,T];L2(Ω))⋂C0((0,T];HN2(Ω)),w∈C0([0,T];W1,2(Ω))⋂C0((0,T];HN3(Ω)).
Here we denote that HNs={u∈Hs(Ω);∂u/∂n=0,on∂Ω}. The proof of Theorem 10 is completed.


## 4. Global Solution and Some A Priori Estimates

In this section, the global-in-time existence of a solution to system (1) is proved. The following a priori estimates will play a crucial role in the proof of our result.Lemma 11.
Suppose that0≤u0∈L2(Ω), (u,v) is a solution to (1), and then there exists a positive constant c0 such that
(28)∥u(t)∥1≤c0,∀t>0.Proof.
Integrating the first equation of (1) on Ω, we obtain
(29)d∥u(t)∥1dt=∫Ω‍(au-bu2)dx=a∥u∥1-b∫Ω‍u2dx≤a∥u∥1-b|Ω|∥u∥12.
From the above equation, it is easy to know that d∥u(t)∥1/dt<0 if ∥u(t)∥1≥a|Ω|/b. Then there exists a constant c0 (depending on a, b, and Ω) such that ∥u(t)∥1≤c0.Lemma 12.
Suppose that0≤u0∈L2(Ω), 0≤w0∈W1,2(Ω), and α<1/n; (u,w) is a local solution to (1) in [0,T] satisfying
(30)u∈C0([0,T];L2(Ω))⋂C0((0,T];H1(Ω))⋂C0((0,T];HN2(Ω)),w∈C0([0,T];W1,2(Ω))⋂C0((0,T];HN2(Ω))⋂C0((0,T];HN3(Ω)).
Then for any τ>0,
(31)∥u(t)∥L22+∥w(t)∥W1,22≤c(1+e-υt(∥u0∥L22+∥w0∥W1,22)),0≤t≤T,∥u(t)∥H1+∥w(t)∥H2≤C,τ<t≤T,
where C depends only on Ω, uτ, wτ, and υ>0.Proof.
In the process of the proof, we denote any positive constant byc which may change from line to line and let τ be a small enough constant.
Step  1. Taking the inner product of the first equation of (1) with u in L2(Ω),
(32)12d∥u(t)∥L22dt+∫Ω‍|∇u|2dx=∫Ω‍uα∇u·∇g(w)dx+∫Ω‍u(au-bu2)dx≤12∫Ω‍|∇u|2dx+12∫Ω‍u2α|∇w|2dx+a24b∫Ω‍udx.
By Hölder’s inequality,
(33)∫Ω‍u2α|∇w|2dx≤(∫Ω‍u2pαdx)1/p(∫Ω‍|∇w|2p′dx)1/p′=∥u∥L2pα2α∥∇w∥L2p′2(1p+1p′=1).
If 2pα(n-2)≤2n and 2pα>1, then, from Lemma 3, there exists a=(n-(n/2pα))/(1+(n/2))∈(0,1) such that
(34)∥u∥L2pα2α≤∥u∥W1,22aα∥u∥L12α(1-a).
Lemmas 9 and 11 imply that ∥∇w(t)∥q is bounded for any q∈(1,n/(n-1)).
Ifq<n/(n-1)<2p′<2n/(n-2), then there exists b=((n/q)-(n/2p′))/(1-(n/2)+(n/q))∈(0,1) such that
(35)∥∇w∥L2p′2≤c∥∇w∥W1,22b∥∇w∥Lq2(1-b)≤c∥▵w∥L22b.
Lett=1/p′>(n-2)/n,g(α,p,q)=2aα+2b since α<1/n,
(36)g(α,p,n-1)=4nα-2n(1-t)n+2+2n(2n-2-nt)=(4α+2)n2+4n-8n(n+2)+-4tn+2<(4α+2)n2n(n+2)<2.
Then there exist p, q which satisfy the above condition such that 2aα+2b<2. By Lemma 6, for any ε>0, there is a constant cε such that
(37)∫Ω‍u2α|∇w|2dx≤∥∇u(t)∥L22aα∥▵w(t)∥L22b≤ε(∥∇u(t)∥L22+∥▵w(t)∥L22)+cε.
Taking the inner product of the second equation of (1) with ▵w in L2(Ω), for any ε1>0,
(38)12d∥∇w(t)∥L22dt+∫Ω‍|▵w|2dx+∫Ω‍|∇w|2dx=-∫Ω‍u▵wdx≤ε122∫Ω‍|∇u|2dx+12ε12∫Ω‍|∇w|2dx.
From (32) and (38), there are ε, ε1 such that υ=min{1-ε-ε12,2-1/ε12}>0 and
(39)d(∥∇w(t)∥L22+∥u(t)∥L22)dt+υ(∫Ω‍|▵w|2dx+∫Ω‍|∇w|2dx+∫Ω‍|∇u|2dx)≤c.
By the Poincaré inequality,
(40)∫Ω‍|∇u|2dx≥c∫Ω‍u2dx,∫Ω‍|▵w|2dx≥c∫Ω‍|∇w|2dx.
Gronwall's lemma implies that
(41)∥u(t)∥L22+∥w(t)∥W1,22≤c(1+e-υt(∥u0∥L22+∥w0∥W1,22)),t∈[0,T].
Step  2. From the analysis in Step 1, ∥u(t)∥L2 is bounded in [0,T]. Furthermore, by using Lemma 9, for any p<2n/(n-2) and small enough constant τ>0, there is a constant c such that ∥∇w(t)∥Lp≤c for all t∈[τ,T] since w(x,t)∈C2,1(Ω;(τ,T])∩C((τ,T];C3(Ω)). Applying the operator ∇ on both sides of the second equation of (1),
(42)∂∇w∂t=∇(▵w)-∇w+∇u.
Taking the inner product of (42) with -∇(∆w), for any ε2>0,
(43)12d∥▵w∥L22dt+∫Ω‍|∇(▵w)|2dx+∫Ω‍|▵w|2dx=-∫Ω‍∇u·∇(▵w)dx≤ε222∫Ω‍|∇(▵w)|2dx+12ε22∫Ω‍|∇u|2dx.
From (32) and (43), there exist ε=1/4, ε2=2, and a constant cε,ε2 depending only on a,b,|Ω| such that
(44)12(d∥u(t)∥L22dt+d∥▵w∥L22dt)+18∥∇u∥L22+78∥▵w∥L22≤cε,ε2.
By Poincaré inequality and Gronwall's lemma, then
(45)∥u∥L22+∥▵w∥L22≤c(2cε,ε2+e(-1/8)t(∥uτ∥L22+∥▵wτ∥L22)),t∈(τ,T].
Taking the inner product of the first equation of (1) with -▵u in L2(Ω), then
(46)12d∥∇u∥L22dt+∫Ω‍|▵u|2dx=∫Ω‍∇·(uα∇g(w))▵udx+∫Ω‍f′(u)|∇u|2dx≤ε322∫Ω‍|▵u|2dx+12ε32∫Ω‍|∇·(uα∇g(w))|2dx+a∫Ω‍|∇u|2dx.
Since g(s) is a real smooth function and |g′(s)|<λ for all s∈[0,+∞), then there is a constant M>0, such that |g′′(s)|≤M and
(47)|∇·((u+1)α∇g(w))|2=|αg′(w)(u+1)α-1∇u·∇w+g′′(w)(u+1)α|∇w|2+g′(w)(u+1)α▵w|2≤c[|∇u·∇w|2+(u+1)2α|▵w|2+(u+1)2α|∇w|4].
By Hölder’s inequality,
(48)∫Ω‍|∇u·∇w|2dx≤∥∇u∥L2p2∥∇w∥L2q2,
where 1/p+1/q=1. There is p∈(n/2,n/(n-2)) for n=2,3 such that 2q<2n/(n-2), 2p(n-2)<2n. By using Lemma 9, there is a constant c such that ∥∇w∥L4≤c. Then from Lemmas 3 and 7 and the Poincaré inequality, for any ε0>0, there exists a constant cε0 such that
(49)∥∇u∥L2p2≤∥∇u∥W1,22a∥∇u∥L22(1-a)≤c∥u∥H22a∥u∥H12(1-a)≤c∥u∥H22a(∥u∥H21/2∥u∥L21/2)2(1-a)≤ε0∥u∥H22+cε0∥u∥L22,
where a=n/2-n/2p<1.
Using the same method as the above analysis, for anyp1 satisfying p1>n/2, 2p1α≤2, and p1′<n/(n-2), there exist a=n/2-n/2p1∈(0,1) and arbitrary constant δ>0 such that
(50)∫Ω‍(u+1)2α|▵w|2dx≤∥u+1∥L2p1α2α∥▵w∥L2p1′2,∥▵w∥L2p1′2≤∥▵w∥H12a∥▵w∥L22(1-a)≤δ∥∇▵w∥L22+cδ∥w∥H22.
Equation (45) implies that ∥▵w∥L2 and ∥u∥L2 are bounded, and then there exists a constant c such that ∥u+1∥L2pα2α≤∥u+1∥L22α≤c and ∫Ω‍(u+1)2α|▵w|2dx≤cδ∥∇▵w∥L22+ccδ∥w∥H22.
By Lemma9 and α<1/n, for any p2 satisfying p2>n/(4-n), 2p2α≤2, and 4p2′<2n/(n-2), there exists a constant c such that
(51)∫Ω‍(u+1)2α|∇w|4dx≤∥u+1∥L2p2α2α∥∇w∥L4p2′4≤∥u+1∥L22∥∇w∥L4p2′4≤c,
where 1/p1+1/p1′=1.
From the above analysis, ifε2=1, ε3=1 in (43), (46) and ε0, δ are small enough, then there exists positive constant υ>0 such that
(52)12d(∥▵w∥L22+∥∇u∥L22)dt+∫Ω‍|∇(▵w)|2dx+υ∫Ω‍(|▵w|2+|∇u|2)dx≤c(∥u∥L2,∥w∥H1).
By Gronwall's lemma,
(53)∥w∥H2+∥u∥H1≤c[c(ε,ε1)+e-υt(∥wτ∥H2+∥uτ∥H1)],t∈(τ,T].
Equations (45), (53), and the choice of T (in Theorem 10) depending on ∥u0∥L2(Ω)+∥w0∥W1,2(Ω) imply that (u(x,T),w(x,T))∈S. It is clear by a standard argument that the solution (u,w) to (1) can be extended up to some Tmax≤∞. With the same method as in the proof of Lemma 12, for any finite Tmax,
(54)∥u(Tmax)∥L22+∥w(Tmax)∥W1,22≤c(1+e-υTmax(∥u0∥L22+∥w0∥W1,22))∥u(Tmax)∥H1+∥w(Tmax)∥H2≤C
which implies that Tmax=+∞. The global existence of the solution to (1) is obtained as the following theorem.Theorem 13.
Suppose that nonnegative functionsu0∈L2(Ω), w0∈W1,2(Ω), and α<1/n, and then there is a unique nonnegative global solution (u,w) to (1) satisfying
(55)u∈C([0,∞);L2(Ω))⋂C1((0,∞);L2(Ω))⋂C((0,∞);H1(Ω)),w∈C([0,∞);W1,2(Ω))⋂C1((0,∞);W1,2(Ω))⋂C((0,∞);H2(Ω)),∥u(t)∥L22+∥w(t)∥W1,22≤c(1+e-υt(∥u0∥L22+∥w0∥W1,22)),t≥0,∥u(t)∥H1+∥w(t)∥H2≤C,t>T1+1,
where C depends only on Ω,uT1,wT1, and T1=(1/υ)ln(∥u0∥L22+∥w0∥W1,22).Remark 14.
From the estimates in Lemma12, there exists fixed constant M>0 and T1=(1/υ)ln(∥u0∥L22+∥w0∥W1,22) such that
(56)∥u(t)∥L2+∥w(t)∥W1,2≤M,t>T1.
Denote the set B={(u,w)∈L2(Ω)×H1(Ω):∥u,w∥L2×H1≤M;u≥0,w≥0}, where M is the constant in (56). The results of Theorem 13 imply that the existence of a dynamical system {S(t)}t≥0 which maps V={(u,w):(u,w)∈L2×H1;u≥0,w≥0} into itself and satisfying (u(t),w(t))=S(t)(u0,w0). Since B is bounded, by Lemma 12, there exists T(B) depending only on B and |Ω| such that
(57)S(t)B⊂B,t≥T(B)
which implies that B is a bounded absorbing set of the semigroup {S(t)}t≥0.
Next, by the Sobolev embedding theorem, the asymptotical compactness of the semigroup{S(t)}t≥0 is shown, and then the existence of a global attractor to system (1) is given.Theorem 15.
Assume thatα<1/n. Then the problem (1) has a global attractor which is a compact invariant set and attracts every bounded set in V.Proof.
If(un,wn) is bounded in V, assume that there exists R such that ∥(un,wn)∥V≤R. Then by Lemma 12, there is a constant T1(R) (depending on R) such that
(58)(un(t),wn(t))=S(t)(un,wn)⊂B,t≥T1(R),
where B is the absorbing set given in (57) and n=1,2,…. For any sequence tn (tn→+∞ as n→∞), there exists N such that, for any n>N(R), tn≥T1(R)+1,
(59)(un(tn),wn(tn))=S(tn-T1(R)-1)(S(T1(R)+1)(un,wn)).
Since the two embeddingH1(Ω)↪L2(Ω),H2(Ω)↪H1(Ω) are compact, from the estimates in Theorem 13, it is clear that {un(tn)} lies in a compact set in L2(Ω) and {wn(tn)} lies in a compact set in H1(Ω). Hence, {S(tn)(un,wn)|Ω} is precompact in L2(Ω)×H1(Ω), which implies that {S(t)}t≥0 is asymptotically compact. In Remark 14, we have obtained the bounded absorbing set of {S(t)}t≥0. Then by Proposition 1, we obtain the existence of the global attractor to (1).



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*Source: 536381-2013-07-10.xml*