# Asymptotic Behavior of Approximated Solutions to Parabolic Equations with Irregular Data

Authors: Weisheng Niu; Hongtao Li
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

LetΩ be a smooth bounded domain in ℝN,(N≥3). We consider the asymptotic behavior of solutions to the following problem ut-div(a(x)∇u)+λf(u)=μinΩ×ℝ+,u=0on∂Ω×ℝ+,u(x,0)=u0(x)inΩ, where u0∈L1(Ω), μ is a finite Radon measure independent of time. We provide the existence and uniqueness results on the approximated solutions. Then we establish some regularity results on the solutions and consider the long-time behavior.


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## Body

## 1. Introduction

We consider the asymptotic behavior of solutions to the following equations(1.1)ut-div(a(x)∇u)+λf(u)=μ,inΩ×ℝ+,u=0,on⁡∂Ω×R+,u(x,0)=u0(x),inΩ,
where Ω is a bounded domain in ℝN(N≥2) with smooth boundary ∂Ω, u0∈L1(Ω), λ≥0,μ is a finite Radon measure independent of time, a(x) is a matrix with bounded, measurable entries, and satisfying the ellipticity assumption
(1.2)a(x)ξ⋅ξ≥c|ξ|2,foranyξ∈RN,withc>0.
Concerning the nonlinear term, we assume that f is a C1 function satisfying,
(1.3)f′(s)≥-l,(1.4)|f′(s)|≤C(1+|s|p-2),p≥2,(1.5)C|s|p-k≤f(s)s≤C|s|p+k,p≥2,
both for all s∈R, where l,C,k are positive constants.Parabolic equations withL1 or measure data arise in many physical models, control problems, and in models of turbulent flows in oceanography and climatology [1–4]. Existence and regularity results for parabolic equations with L1 and measure data have been studied widely by many authors in the past decades, see [5–10]. The usual approach to study problems with these kinds of data is approximation. The basic reference for these arguments is [7], where the authors obtained weak solutions (in distribution sense) to nonlinear parabolic equations. In our setting, such a solution is a function u∈L1((0,T);W01,1(Ω)) such that f(u)∈L1(QT) for any T>0, and
(1.6)-∫QT‍uφtdxdt+∫QT‍a(x)∇u∇φdxdt+λ∫QT‍f(u)φdxdt=∫QT‍φdμ+∫Ω‍u0(x)φ(0,x)dx,
for any φ∈Cc∞([0,T)×Ω).Generally, the regularity of weak solutions (in distribution sense) is not strong enough to ensure uniqueness [8]. But one may select a weak solution which is “better” than the others. Since one may prove that the weak solution obtained from approximation does not depend on the approximation chosen for the irregular data. In such a sense, it is the only weak solution which is found by means of approximations; we may call it approximated solutions. Such a concept was first introduced by [9]. Here in the present paper, we will focus ourselves to the scope of approximated solutions, that is, weak solutions obtained as limits of approximations.The long-time behavior of parabolic problems with irregular data (such asL1 data, measure data) have been considered by many authors [11–16]. In [11, 12], existence of global attractors for porous media equations and m-Laplacian equations with irregular initial data were deeply studied, while in [13, 14] the convergence to the equilibrium for the solutions of parabolic problems with measrued data were thoroughly investigated. In [15, 16], we considered the existence of global attractors for the parabolic equations with L1 data.In this paper, we intend to consider the asymptotic behavior of approximated solutions to problem (1.1) with measure data. Precisely speaking, we assume that the forcing term in the equations is just a finite Radon measure. For the case λ>0, to ensure the existence result for large p in (1.5) [17], we restrict ourselves to diffuse measures, that is, μ does not charge the sets of zero parabolic 2-capacity (see details for parabolic p-capacity in [18]). We first provide the existence result for problem (1.1) and prove the uniqueness of the approximated solution. Then using some decomposition techniques, we establish some new regularity results and show the existence of a global attractor 𝒜 in Lp-1(Ω)∩W01,q(Ω) with q<max{N/(N-1),(2p-2)/p}, which attracts every bounded subset of L1(Ω) in the norm of Lr(Ω)∩H01(Ω), for any r∈[1,∞).For the caseλ=0, we consider general bounded Radon measure μ which is independent of time. We provide the uniqueness of approximated solutions for the parabolic problem and its corresponding elliptic problem. Then we prove that the approximated solution of the parabolic equations converges to the unique approximated solution of the corresponding elliptic equations in the norm topology of Lr(Ω)∩H01(Ω), for any r∈[1,∞), though they all lie in some less regular spaces.Our main results can be stated as follows.Theorem 1.1.
Assume thatu0∈L1(Ω), λ>0, μ is a bounded Radon measure, which does not charge the sets of zero parabolic 2-capacity and is independent of time, f is a C1 function satisfying assumptions (1.3)–(1.5). Then the semigroup {S(t)}t≥0, generated by approximated solutions of problem (1.1), possesses a global attractor 𝒜 in L1(Ω). Moreover, 𝒜 is compact and invariant in Lp-1(Ω)∩W01,q(Ω) with q<max{N/(N-1),(2p-2)/p}, and attracts every bounded subset of L1(Ω) in the norm topology of Lr(Ω)∩H01(Ω), 1≤r<∞.Theorem 1.2.
Assume thatu0∈L1(Ω), λ=0, μ is a bounded Radon measure independent of time. Then the approximated solution u(t) of problem (1.1) is unique and converges to the unique approximated solution of the corresponding elliptic equations in the norm topology of Lr(Ω)∩H01(Ω), for any 1≤r<∞.Remark 1.3.
Thoughu(t) and v all lie in some less-regular spaces, u(t) converges to v in stronger norm, that is, u(t)-v converges to 0 in Lr(Ω)∩H01(Ω), 1≤r<∞. Such a result, in some sense, sharpens the result of [13], where the author showed that u(t) converges to v in L1(Ω).We organize the paper as follows: in Section2, we provide the existence of approximated solutions, prove the uniqueness result and some useful lemmas; in Section 3, we establish some improved regularity results on the approximated solutions. At last, in Section 4, we prove the main theorems.For convenience, for anyT>0 we use QT to denote Ω×(0,T) hereafter. Also, we denote by |E| the Lebesgue measure of the set E, and denote by C any positive constant which may be different from each other even in the same line.


## 2. Existence Results and Useful Lemmas

We begin this section by providing some existence results on the approximated solutions.Definition 2.1.
A functionu is called an approximated solution of problem (1.1), if u∈L1((0,T);W01,1(Ω)), f(u)∈L1(QT) for any T>0, and
(2.1)-∫QT‍uφtdxdt+∫QT‍a(x)∇u∇φdxdt+λ∫QT‍f(u)φdxdt=∫QT‍φdμ+∫Ω‍u0(x)φ(0,x)dx,
for any φ∈Cc∞([0,T)×Ω), and moreover, u is obtained as limit of solutions to the following approximated problem
(2.2)utn-div(a(x)∇un)+λf(un)=μn,inΩ×ℝ+,u=0,on⁡∂Ω×R+,u(x,0)=u0n,inΩ,
where {μn},{u0n} is a smooth approximation of data μ,u0.Theorem 2.2.
Under the assumptions of Theorem1.1, problem (1.1) has a unique approximated solution u∈C([0,T];L1(Ω)), for any T>0, satisfying  (i)
ifp≥(2N+2)/N, then u∈Lq((0,T);W01,q(Ω)) with q<(2p-2)/p;(ii)
if2≤p<(2N+2)/N, then u∈Lq((0,T);W01,q(Ω)) with q<(N+2)/(N+1).Proof.
According to [18, Theorem 2.12], if a Radon measure μ on QT does not charge the sets of zero parabolic 2-capacity and is independent of time, μ can actually be identified as a Radon measure which is absolutely continuous with respect to the elliptic 2-capacity. Using Theorem 2.1 of [19], μ can be decomposed as μ=g+divG, where g∈L1(Ω),G∈(L2(Ω))N. Hence, we need only to consider the following problem
(2.3)ut-div(a(x)∇u)+λf(u)=g+divG,inΩ×ℝ+,u=0,on⁡∂Ω×R+,u(x,0)=u0,inΩ.
The proof of existence part of the theorem is similar to [9]. Besides, one can prove u∈C([0,T];L1(Ω)) using arguments similar to CLAIM 2 in [8]. So we omit the details of them and only prove the uniqueness result.
Let{gn}n∈ℕ, {u0n}n∈ℕ be a smooth approximation of data g and u0 with
(2.4)‖u0n‖L1(Ω)≤‖u0‖L1(Ω),‖g~n‖L1(Ω)≤‖g‖L1(Ω),
and let {g~n}n∈ℕ, {u~0n}n∈ℕ be another smooth approximation of the data with
(2.5)‖u~0n‖L1(Ω)≤‖u0‖L1(Ω),‖g~n‖L1(Ω)≤‖g‖L1(Ω).
Assume that u,u~ are two approximated solutions to problem (1.1), obtained as limit of the solutions to the following two approximated problems, respectively,
(2.6)utn-div(a(x)∇un)+λf(un)=gn+divG,inΩ×ℝ+,u=0,on⁡∂Ω×R+,u(x,0)=u0n,inΩ,(2.7)u~tn-div(a(x)∇u~n)+λf(u~n)=g~n+divG,inΩ×ℝ+,u=0,on⁡∂Ω×R+,u(x,0)=u~0n,inΩ.
Now we prove that u=u~. For any k>0, define ψk(s) as
(2.8)ψk(s)={k,s>ks,|s|≤k-k,s<-k.
Let Ψk(σ)=∫0σ‍ψk(s)ds be its primitive function. Taking ψk(un-u~n) as a test function in (2.6) and (2.7), we deduce that
(2.9)∫Ω‍Ψk(un-u~n)(t)dx-∫Ω‍Ψk(u0n-u~0n)dx+∫QT‍a(x)(∇ψk(un-u~n))2dxdt=λ∫QT‍(f(u~n)-f(un))ψk(un-u~n)dxdt+∫QT‍(gn-g~n)ψk(un-u~n)dxdt≤∫QT‍λl(un-u~n)ψk(un-u~n)dxdt+∫QT‍|gn-g~n|dxdt.
Hence, from the assumptions on f, we get
(2.10)∫Ω‍Ψk(un-u~n)(t)dx≤2lλ∫0T‍∫Ω‍Ψk(un-u~n)dxdt+∫QT‍|gn-g~n|dxdt+∫Ω‍Ψk(u0n-u~0n)dx.
Let n→∞, we have
(2.11)∫Ω‍Ψk(u-u~)(t)dx≤2lλ∫0T‍∫Ω‍Ψk(u-u~)(t)dxdt.
Thus for all k>0, we have
(2.12)sup⁡[0,T]∫Ω‍Ψk(u~-u)(t)dx≤2lTsup⁡[0,T]∫Ω‍Ψk(u~-u)dx.
Taking T′ small enough such that 2lλT′<1, we deduce that Ψk(u¯-u)=0 for all k>0 in QT′, thus u≡u¯ in QT′. Dividing [0,T] into several intervals to carry out the same arguments, we obtain the uniqueness of the approximated solution.
Similar to [20], we can prove the following.Theorem 2.3.
Under the assumptions of Theorem1.1, there exists at least one approximated solution v to the stationary problem of corresponding to problem (1.1), with  (i)
ifp≥(2N-2)/(N-2), we have v∈W01,q(Ω) for q<(2p-2)/p;(ii)
if2≤p<(2N-2)/(N-2), we have v∈W01,q(Ω) for q<N/(N-1).Remark 2.4.
Note that ifv is an approximated solution to following problem
(2.13)-div(a(x)∇v)+λf(v)=g+divG,inΩ,v=0,on⁡∂Ω,
then there is a sequence {vn} converges to v, where vn is the solution of the corresponding approximated problem
(2.14)-div(a(x)∇vn)+λf(vn)=gn+divG,inΩ,vn=0,on⁡∂Ω.
And hence vn is a solution of parabolic equations
(2.15)vtn-div(a(x)∇vn)+λf(vn)=gn+divG,inΩ×ℝ+,vn=0,on⁡∂Ω×R+,vn(0)=vn(x),inΩ.
Thus, v is an approximated solution of problem (1.1) with initial data u0=v(x).Under the assumptions of Theorem1.2, the problem turns out to be
(2.16)ut-div(a(x)∇u)=μ,inΩ×ℝ+,u=0,on⁡∂Ω×R+,u(x,0)=u0,inΩ.
The existence of approximated solutions to problem (1.1) and the elliptic equations corresponding to it follows directly from Sections IV and II of [7]. Form Section 2.3 of [21], we know that the approximated solution to the stationary equations is actually a duality solution, and hence unique. Furthermore, it is not difficult to prove that an approximated solution for linear parabolic equations turns out to be a duality solution, and hence unique too.Lemma 2.5.
Under the assumptions of Theorem1.2, an approximated solution to the parabolic problem (1.1), (2.16) turns out to be a duality solution, and conversely.Proof.
The proof is mainly similar to that of Theorem 6 in [22]. We just sketch it. Let u be an approximated solution, then there exist a smooth approximation {μn}n∈ℕ, {u0n}n∈ℕ of data μ and u0, such that the solution of the approximated problem of (2.16) with data μn and u0n converges to u. Let h∈Cc∞(QT) and ω be the solution of the the following parabolic problem
(2.17)-ωt-div(a*(x)∇ω)=h,inΩ×(0,T),ω=0,on⁡∂Ω×(0,T),ω(x,T)=0,inΩ,
where a*(x) is the transposed matrix of a(x). Taking ω as a test function in the approximated problem and taking un as a test function in the problem above, then let n go to infinity we obtain that the approximated solution is a duality solution. Form the uniqueness of duality solutions [13], we get the conclusion.Now we provide two lemmas which are useful in analyzing the regularity and asymptotic behavior of the solutions to problem (1.1).Lemma 2.6 (see [15]).
LetX, Y be two Banach spaces, let X be separable, reflexive, and let X⊂Y with dual X*. Suppose that {un} is uniformly bounded in L∞((0,T);X) with
(2.18)esssup⁡t∈[0,T]‖un(t)‖X≤C,
and that un→u weakly in Lr((0,T);X) for some r∈(1,∞). Then
(2.19)esssup⁡t∈[0,T]‖u(t)‖X≤C.
Moreover, if u∈C([0,T];Y), then in fact
(2.20)sup⁡t∈[0,T]‖u(t)‖X≤C.Lemma 2.7 (see [16]).
LetX,Y be two Banach spaces with imbedding X↪Y, let {S(t)}t≥0 be a continuous semigroup on Y. Assume that {S(t)}t≥0 is asymptotically compact in X and has an absorbing set B0⊂X, that is, for any bounded set B⊂Y, there exists a T=T(B) such that
(2.21)S(t)B⊂B0,∀t≥T.
Then {S(t)}t≥0 has a global attractor 𝒜 in X, which is compact, invariant in X and attracts every bounded sets of Y in the topology of X.Remark 2.8.
To obtain global attractors, one usually needs the semigroup to be norm-to-norm continuous, weak-to-weak, or norm-to-weak continuous [23–27]. Here, to obtain global attractors in the space X, we need neither of them. We only need the semigroup to be continuous in a less-regular space Y.


## 3. Improved Regularity Results on the Approximated Solutions

In this section, we prove the following regularity results on the approximated solutionu to problem (1.1).Theorem 3.1.
Under the assumptions of Theorem2.2, let u(t) be the approximated solution to problem (1.1). Then u admits the decomposition u(x,t)=w(x,t)+v(x), with v being an approximated solution to problem (2.13), and w being an approximated solution of the following problem
(3.1)wt-div(a(x)∇w)+λf(v+w)-λf(v)=0,inΩ×ℝ+,w=0,on⁡∂Ω×R+,w(x,0)=u0-v,inΩ.
Moreover, we have  (i)
w∈L∞((δ,T);Lq(Ω)) for any 0<δ<T,1≤q<∞. Moreover, there exists a constant Mq and a time tq(u0,g,G) such that ∥w(t)∥Lq(Ω)≤Mqforallt≥tq(u0,g,G).(ii)
w∈L∞((δ,T);H01(Ω)) for any 0<δ<T. Moreover, there exists a constant ρ and a time T0(u0,g,G) such that ∥w(t)∥H01(Ω)≤ρforallt≥T0(u0,g,G).Proof.
We follow the lines of [15, 28]. Let {gn} be a sequence of smooth data which converges to g in L1(Ω) and ∥gn∥L1(Ω)≤∥g∥L1(Ω). Let vn be a solution of the following approximated problem for each n,
(3.2)-div(a(x)∇vn)+λf(vn)=gn+divG,inΩ,vn=0,on⁡∂Ω.
Then vn converges (up to subsequences) to an approximated solution v strongly in L1(Ω), and weakly in W01,q(Ω),1≤q<N/(N-1). Let {un} be a sequence of solutions to the following approximated problem
(3.3)utn-div(a(x)∇un)+λf(un)=gn(x)+divG,inΩ×ℝ+,un(x)=0,on⁡∂Ω×R+,un(x,0)=u0n,inΩ,
where u0n converges to u0 with ∥u0n∥L1(Ω)≤∥u0∥L1(Ω). Similar to [8, 29], we know that
(3.4)un→uweaklyinLq((0,T);W01,q(Ω)),q<N+2N+1,un⟶uinC([0,T];L1(Ω)).
Now let wn(t)=un(t)-vn. Then wn satisfies
(3.5)wtn-div(a(x)∇wn)+λf(vn+wn)-λf(vn)=0,inΩ×ℝ+,wn=0,on⁡∂Ω×R+,wn(x,0)=u0n-vn,inΩ.
Similarly, we have wn (up to subsequences) converges to the approximated solution w of problem (3.1) in C([0,T];L1(Ω)) and weakly in Lq((0,T);W01,q(Ω)),q<(N+2)/(N+1).
Now we prove (i). Takingψ1(un) as test function in (3.3) (for simplicity we take λ=1), we deduce that
(3.6)ddt∫Ω‍Ψ1(un)dx+∫Ω‍f(un)ψ1(un)dx≤‖g‖L1(Ω)+C‖G‖L2(Ω).
Since
(3.7)f(un)ψ1(un)≥(C|un|p-1-C)|ψ1(un)|,
we have
(3.8)ddt∫Ω‍Ψ1(un)dx+C∫Ω‍|Ψ1(un)|dx≤‖g‖L1(Ω)+C‖G‖L2(Ω)+C|Ω|.
The Gronwall's inequality implies that
(3.9)∫Ω‍Ψ1(un(t))dx≤‖u0‖L1(Ω)e-Ct+C|Ω|+C‖g‖L1(Ω)+C‖G‖L2(Ω).
Noticing that
(3.10)∫Ω‍|un(t)|dx≤∫Ω‍Ψ1(un(t))dx+|Ω|,
we obtain that
(3.11)∫Ω‍|un(t)|dx≤‖u0‖L1(Ω)e-Ct+C|Ω|+C‖g‖L1(Ω)+C‖G‖L2(Ω),t≥0.
Moreover, integrating (3.6) between t and t+1 and using (3.7) we have
(3.12)∫tt+1‍∫Ω‍|un|p-1dxdξ≤C(‖u0‖L1(Ω)e-Ct+|Ω|+‖g‖L1(Ω)+‖G‖L2(Ω)).
Similarly, taking ψ1(vn) as test function in (3.2), we can deduce that
(3.13)∫Ω‍|f(vn)|dx≤C∫Ω‍(|vn|p-1+1)dx≤C(‖g‖L1(Ω)+‖G‖L2(Ω)+|Ω|).
Hence,
(3.14)∫tt+1‍∫Ω‍|wn|p-1dxdξ≤C(‖g‖L1(Ω)+‖G‖L2(Ω)+‖u0‖L1(Ω)+|Ω|),
with C independent of n, for t≥0.
Now we use bootstrap method in the casep≥3. The case 2≤p<3 can be treated similarly with minor modifications. Multiplying (3.5) by |wn|q0-2wn,q0=p-1≥2, and integrating on Ω, we obtain
(3.15)1q0ddt∫Ω‍|wn|q0dx+(q0-1)c∫Ω‍|∇wn|2|wn|q0-2dx≤l∫Ω‍|wn|q0dx.
Since |∇wn|2|wn|q0-2=(2/q0)2|∇(|wn|(q0-2)/2wn)|2, we deduce that
(3.16)ddt∫Ω‍|wn|q0dx+∫Ω‍|∇(|wn|(q0-2)/2wn)|2dx≤C∫Ω‍|wn|q0dx.
Integrating (3.16) between s and t+1(t≤s<t+1), it yields
(3.17)∫Ω‍|wn(t+1)|q0dx≤C∫st+1‍∫Ω‍|wn|q0dxdξ+∫Ω‍|wn(s)|q0dx.
Integrating the above inequality with respect to s between t and t+1, we get
(3.18)∫Ω‍|wn(t+1)|q0dx≤C∫tt+1‍∫Ω‍|wn|q0dxdξ.
Therefore,
(3.19)∫Ω‍|wn(t)|q0dx≤C,∀t≥1.
Integrating (3.16) on [t,t+1] for t≥1, we deduce that
(3.20)∫tt+1‍∫Ω‍|∇(|wn|(q0-2)/2wn)|2dxdξ≤C∫tt+1‍∫Ω‍|wn(ξ)|q0dxdξ+C∫Ω‍|wn(t)|q0dx≤C.
Note that (3.20) insures that, for any t≥1, there exists at least a t0∈[t,t+1] such that
(3.21)∫Ω‍|∇(|wn(t0)|(q0-2)/2wn(t0))|2dx≤C.
Standard Sobolev imbedding implies that
(3.22)∫Ω‍|wn(t0)|q0(N/(N-2))dx≤C.
Now multiplying (3.5) by |wn|q1-2wn,q1=(N/(N-2))q0, we have
(3.23)N-2Nq0ddt∫Ω‍|wn|(N/(N-2))q0dx+C(c,q0,N)∫Ω‍|∇(|wn|(Nq0-2N+4)/(2N-4)wn)|2dx≤l∫Ω‍|wn|(N/(N-2))q0dx.
Using Hölder inequality, and Young inequality we deduce that
(3.24)∫Ω‍|wn|(N/(N-2))q0dx≤ε′∫Ω‍|∇(|wn|(Nq0-2N+4)/(2N-4)wn)|2dx+Cε(∫Ω‍|wn|q0dx)N/(N-2).
Taking (3.24) into (3.23), it yields
(3.25)ddt∫Ω‍|wn|(N/(N-2))q0dx+C′(c,q0,N)∫Ω‍|∇(|wn|(Nq0-2N+4)/(2N-4)wn)|2dx≤Cε′(∫Ω‍|wn|q0dx)N/(N-2).
Integrating (3.25) between t0 and t0+s,0<s≤1, we have
(3.26)∫Ω‍|wn(t0+s)|(N/(N-2))q0dx≤∫Ω‍|wn(t0)|(N/(N-2))q0dx+Cε′(∫Ω‍|wn|q0dx)N/(N-2).
Therefore, from (3.19) and (3.22) we get
(3.27)∫Ω‍|wn(t)|(N/(N-2))q0dx=∫Ω‍|wn(t)|q1dx≤C,∀t≥2,
with C independent of n. Integrating (3.25) between t and t+1 for t≥2, we obtain
(3.28)∫tt+1‍∫Ω‍|∇(|wn|(q1-2)/2wn)|2dxdξ=∫tt+1‍∫Ω‍|∇(|wn|(Nq0-2N+4)/(2N-4)wn)|2dxdξ≤C.
Similar to (3.22), for any t≥2, there exists at least a t0∈[t,t+1] such that
(3.29)∫Ω‍|wn(t0)|q0(N2/(N-2)2)dx≤C.
Bootstrap the above processes, we can deduce that
(3.30)∫Ω‍|wn(t)|qkdx≤C,∀t≥Tk,
with qk=(N/(N-2))kq0, and C independent of n. Note that wn→w in C([0,T];L1(Ω)) and w∈C([0,T];L1(Ω)). From Lemma 2.6, we have
(3.31)‖w(t)‖Lqk(Ω)qk=∫Ω‍|w(t)|qkdx≤C,∀t≥Tk.
Taking k large enough, we get the second part of (i) proved. If the integration are taken over [t,t+δ0] instead of [t,t+1], we get the first part of (i).
Now we are in the position to prove (ii). We multiply (3.5) with wn and deduce that
(3.32)12ddt∫Ω‍|wn|2dx+c∫Ω‍|∇wn|2dx≤l∫Ω‍|wn|2dx,
integrating over [t,t+1],t≥T′, we get
(3.33)∫tt+1‍∫Ω‍|∇wn|2dxdt≤C,
with C independent of n. Now, multiplying (3.5) with wtn, we obtain
(3.34)∫Ω‍|wtn|2dx+ddt∫Ω‍a(x)|∇wn|2dx+ddt∫Ω‍(F(wn+vn)-F(vn)-f(vn)wn)dx=0,
where F(v+σ)=∫0σ‍f(v+s)ds. Integrating (3.34) between s and t+1(t≤s<t+1) gives
(3.35)∫Ω‍|∇wn(t+1)|2dx+∫Ω‍(F(wn(t+1)+vn)-F(vn)-f(vn)wn(t+1))dx≤C∫Ω‍|∇wn(s)|2dx+C∫Ω‍(F(wn(s)+vn)-F(vn)-f(vn)wn(s))dx.
Now, integrating the above inequality with respect to s between t and t+1 we have
(3.36)∫Ω‍|∇wn(t+1)|2dx+∫Ω‍(F(wn(t+1)+vn)-F(vn)-f(vn)wn(t+1))dx≤C∫tt+1‍∫Ω‍|∇wn|2dxdξ+C∫tt+1‍|∫Ω‍(F(wn+vn)-F(vn)-f(vn)wn)dx|dξ.
Since
(3.37)∫Ω‍(F(wn+vn)-F(vn)-f(vn)wn)dx=∫Ω‍∫01‍(f(vn+swn)-f(vn))wndsdx,f(vn+τwn)-f(vn)=∫01‍f′(vn+θτwn)τwndθ,0≤τ≤1.
We deduce that
(3.38)|∫Ω‍(F(wn+vn)-F(vn)-f(vn)wn)dx|≤∫Ω‍|f′(vn+τ′wn)||wn|2dx,0≤τ′≤1.
From the assumption (1.4) on f, we have
(3.39)|∫Ω‍(F(wn+vn)-F(vn)-f(vn)wn)dx|≤∫Ω‍C(|vn|p-2+|wn|p-2+1)|wn|2dx≤C(‖vn‖Lp-1(Ω)p-2+‖wn‖L2p-2(Ω)2+‖wn‖Lp(Ω)p+1).
Using results in (3.13) and (3.30), we know that
(3.40)|∫Ω‍(F(wn+vn)-F(vn)-f(vn)wn)dx|≤C,∀t≥T′′.
Set T0=max{T′,T′′}. Combining (3.33), (3.34), (3.36), and (3.40) we have
(3.41)∫Ω‍|∇wn(t)|2dx≤C,∀t≥T0+1,(3.42)∫tt+1‍∫Ω‍|wtn|2dxdξ≤C,∀t≥T0+1.
From Lemma 2.6, we obtain
(3.43)‖w‖H01(Ω)≤C,∀t≥T0+1.
Thus we get the second part of (ii) proved. Taking integration over [t,t+δ0] instead of [t,t+1], the first part of (ii) follows. The proof is completed now.


## 4. Proof of the Main Theorems

Let{S(t)}t≥0 be the semigroup generated by problem (1.1) and let v(x) be an approximated solution to problem (2.3). Define
(4.1)S1(t)(u0-v(x))=S(t)u0-v(x).
Then it is easy to verify that {S1(t)}t≥0 is a continuous semigroup in (L1(Ω)-v) and hence in L1(Ω). From the results in Section 3, we know that the semigroup {S(t)}t≥0 possesses a global attractor 𝒜 in L1(Ω). To verify the second part of Theorem 1.1, we prove the following theorem.Theorem 4.1.
Under the assumptions of Theorem1.1, the semigroup {S1(t)}t≥0 possesses a global attractor 𝒜v in Lr(Ω)∩H01(Ω),1≤r<∞, that is, 𝒜v is compact, invariant in Lr(Ω)∩H01(Ω),1≤r<∞, and attracts every bounded initial set of L1(Ω) in the norm topology of Lr(Ω)∩H01(Ω),1≤r<∞.Proof.
From Theorem3.1, we know that {S(t)}t≥0 possesses an absorbing set B0 in L1(Ω). Also {S1(t)}t≥0 possesses absorbing sets B1(=B0-v), B2, respectively, in L1(Ω) and Lr(Ω)∩H01(Ω) for any 1≤r<∞.
In the next, we prove the asymptotic compactness of{S1(t)}t≥0. Before that we establish the following estimate
(4.2)∫Ω‍|wtn|2dx≤C,fortlargeenough.
Actually, differentiating (3.5) in time and denoting wn~=wtn, we have
(4.3)wn~t-div⁡(a(x)∇wn~)+f′(vn+wn)wn~=0.
Multiplying (4.3) by wn~ and using (1.3), we deduce that
(4.4)ddt∫Ω‍|wn~|2dx≤l∫Ω‍|wn~|2dx.
Integrating the above inequality between s and t+1(t≤s<t+1) gives
(4.5)∫Ω‍|wn~(t+1)|2dx≤2l∫st+1‍∫Ω‍|wn~|2dxdξ+∫Ω‍|wn~(s)|2dx.
Integrating the above inequality with respect to s between t and t+1, using (3.42) we obtain
(4.6)∫Ω‍|wn~(t)|2dx≤(2l+1)∫tt+1‍∫Ω‍|wn~|2dxdξ≤(2l+1)C,∀t≥T0+1.
Now we prove the asymptotic compactness of the semigroup {S1(t)}t≥0, that is, for any sequences {w0,k}⊂B1,tk→∞, sequence {wk(tk)} has convergent subsequences, where wk(t)=S1(t)w0,k. Since {S1(t)}t≥0 is compact in Lr(Ω),1≤r<∞, there is a subsequence {wki(tki)}, which is a Cauchy sequence in Lr(Ω),1≤r<∞. Denoting uki(tki)=wki(tki)+v,ukj(tkj)=wkj(tkj)+v, we deduce that
(4.7)c‖wki(tki)-wkj(tkj)‖H01(Ω)2≤〈div⁡(a(x)∇(wki(tki)+v))-div⁡(a(x)∇(wkj(tkj)+v)),wki(tki)-wkj(tkj)〉=〈∂twki(tki)-∂twkj(tkj)+f(uki(tki))-f(ukj(tkj)),wki(tki)-wkj(tkj)〉≤‖∂twki(tki)-∂twkj(tkj)‖L2(Ω)‖wki(tki)-wkj(tkj)‖L2(Ω)+‖f(wki(tki)+v)-f(wkj(tkj)+v)‖Lσ(Ω)‖wki(tki)-wkj(tkj)‖Lσ′(Ω).
We then conclude form (4.6) that {wki(tki)} is a Cauchy sequence in H01(Ω), and thus {S1(t)}t≥0 is asymptotically compact in Lr(Ω)∩H01(Ω),1≤r<∞.
Using Lemma2.7, we conclude that {S1(t)}t≥0 possesses a global attractor 𝒜v, which is compact, invariant in Lr(Ω)∩H01(Ω), and attracts every bounded initial sets of L1(Ω) in the topology of Lr(Ω)∩H01(Ω).Completion of the Proof of Theorem1.1
Note that(4.8)𝒜v=∩s≥0∪t≥sS1(t)B2¯L1(Ω)=∩s≥0∪t≥sS1(t)B1¯L1(Ω)=∩s≥0∪t≥s(S(t)B0-v)¯L1(Ω),A=∩s≥0∪t≥sS(t)B0¯L1(Ω).
Thus we have
(4.9)A=Av+v.
The above relation between 𝒜 and 𝒜v implies the conclusion of Theorem 1.1 directly.Proof of Theorem1.2.
Letu(t),v be the approximated solution to the parabolic and its corresponding elliptic problem respectively. Since the approximated solution is a duality solution and conversely, we conclude that u(t) converges to v in L1(Ω) as t→∞. Using arguments similar to Section 3, we can prove similar regularity results for w(=u-v) and then prove the asymptotic compactness of the semigroup S1(t) as in Theorem 4.1. Thus, we obtain that w(t) converges to 0 in Lr(Ω),1≤r<∞, as t→∞. Moreover from the asymptotic compactness of the semigroup S1(t), we know that w(t) converges to 0 in H01(Ω) as t→∞. Else, we have a sequence tn→∞, such that C>∥w(tn)∥H01(Ω)≥ϵ>0. Since the semigroup S1(t) is asymptotically compact, there is a subsequence tnj→∞, such that w(tnj) converges to a function χ in H01(Ω) and hence in L1(Ω). Thus χ=0. A contradiction!



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*Source: 312536-2012-09-16.xml*