Asymptotic Behavior of Approximated Solutions to Parabolic Equations with Irregular Data

Abstract

Let Ω be a smooth bounded domain in ℝ^𝑁,(𝑁≥3). We consider the asymptotic behavior of solutions to the following problem 𝑢_𝑡−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.

1. Introduction

We consider the asymptotic behavior of solutions to the following equations 𝑢_𝑡−div(𝑎(𝑥𝑢)=∇0𝑢,)+o𝜆n𝑓𝜕(𝑢Ω)×=ℝ𝜇₊,inΩ×ℝ⁺,𝑢(𝑥,0)=𝑢₀,(𝑥),inΩ,(1.1) where Ω is a bounded domain in ℝ^𝑁(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 𝑎(𝑥)𝜉⋅𝜉≥𝑐||𝜉||²,forany𝜉∈ℝ^N,with𝑐>0.(1.2) Concerning the nonlinear term, we assume that 𝑓 is a 𝐶¹ function satisfying, ||𝑓𝑓^(𝑠)≥−𝑙,(1.3)_𝐶|𝑠|_𝑝(𝑠)||≤𝐶1+|𝑠|^𝑝−2,𝑝≥2,(1.4)−𝑘≤𝑓(𝑠)𝑠≤𝐶|𝑠|^𝑝+𝑘,𝑝≥2,(1.5) both for all 𝑠∈𝑅, where 𝑙,𝐶,𝑘 are positive constants.

Parabolic equations with 𝐿¹ 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);𝑊^01,1(Ω)) such that 𝑓(𝑢)∈𝐿¹(𝑄_𝑇) for any 𝑇>0, and −_𝑄𝑇𝑢𝜑_𝑡𝑑𝑥𝑑𝑡+_𝑄𝑇𝑎(𝑥)∇𝑢∇𝜑𝑑𝑥𝑑𝑡+𝜆_𝑄𝑇𝑓(𝑢)𝜑𝑑𝑥𝑑𝑡=_𝑄𝑇𝜑𝑑𝜇+_Ω𝑢₀(𝑥)𝜑(0,𝑥)𝑑𝑥,(1.6) for any 𝜑∈𝐶^∞𝑐([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 as 𝐿¹ 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 𝐿^𝑝−1(Ω)∩𝑊^01,𝑞(Ω) with 𝑞<max{𝑁/(𝑁−1),(2𝑝−2)/𝑝}, which attracts every bounded subset of 𝐿¹(Ω) in the norm of 𝐿^𝑟(Ω)∩𝐻¹⁰(Ω), for any 𝑟∈[1,∞).

For the case 𝜆=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 𝐿^𝑟(Ω)∩𝐻¹⁰(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 that 𝑢₀∈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 𝐿^𝑝−1(Ω)∩𝑊^01,𝑞(Ω) with 𝑞<max{𝑁/(𝑁−1),(2𝑝−2)/𝑝}, and attracts every bounded subset of 𝐿¹(Ω) in the norm topology of 𝐿^𝑟(Ω)∩𝐻¹⁰(Ω), 1≤𝑟<∞.

Theorem 1.2. Assume that 𝑢₀∈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 𝐿^𝑟(Ω)∩𝐻¹⁰(Ω), for any 1≤𝑟<∞.

Remark 1.3. Though 𝑢(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 𝐿^𝑟(Ω)∩𝐻¹⁰(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 Section 2, 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 any 𝑇>0 we use 𝑄_𝑇 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 function 𝑢 is called an approximated solution of problem (1.1), if 𝑢∈𝐿¹((0,𝑇);𝑊^01,1(Ω)), 𝑓(𝑢)∈𝐿¹(𝑄_𝑇) for any 𝑇>0, and −_𝑄𝑇𝑢𝜑_𝑡𝑑𝑥𝑑𝑡+_𝑄𝑇𝑎(𝑥)∇𝑢∇𝜑𝑑𝑥𝑑𝑡+𝜆_𝑄𝑇𝑓(𝑢)𝜑𝑑𝑥𝑑𝑡=_𝑄𝑇𝜑𝑑𝜇+_Ω𝑢₀(𝑥)𝜑(0,𝑥)𝑑𝑥,(2.1) for any 𝜑∈𝐶^∞𝑐([0,𝑇)×Ω), and moreover, 𝑢 is obtained as limit of solutions to the following approximated problem 𝑢^𝑛𝑡−div(𝑎(𝑥𝑢)∇=𝑢0^𝑛),+o𝜆n𝑓𝜕(𝑢Ω^𝑛)×=ℝ𝜇^𝑛₊,inΩ×ℝ⁺,𝑢(𝑥,0)=𝑢_𝑛0,,inΩ,(2.2) where {𝜇^𝑛},{𝑢^𝑛0} is a smooth approximation of data 𝜇,𝑢₀.

Theorem 2.2. Under the assumptions of Theorem 1.1, problem (1.1) has a unique approximated solution 𝑢∈𝐶([0,𝑇];𝐿¹(Ω)), for any 𝑇>0, satisfying (i) if 𝑝≥(2𝑁+2)/𝑁, then 𝑢∈𝐿^𝑞((0,𝑇);𝑊^01,𝑞(Ω)) with 𝑞<(2𝑝−2)/𝑝;(ii) if 2≤𝑝<(2𝑁+2)/𝑁, then 𝑢∈𝐿^𝑞((0,𝑇);𝑊^01,𝑞(Ω)) with 𝑞<(𝑁+2)/(𝑁+1).

Proof. According to [18, Theorem 2.12], if a Radon measure 𝜇 on 𝑄_𝑇 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 𝑢_𝑡−div(𝑎(𝑥)∇𝑢𝑢)=+0𝜆,𝑓(o𝑢n)=𝜕Ω𝑔×+ℝdi₊v𝐺,inΩ×ℝ⁺,𝑢(𝑥,0)=𝑢₀,,inΩ.(2.3) 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 {𝑔^𝑛}_𝑛∈ℕ, {𝑢^𝑛0}_𝑛∈ℕ be a smooth approximation of data 𝑔 and 𝑢₀ with ‖‖𝑢^𝑛0‖‖_𝐿1(Ω)≤‖‖𝑢₀‖‖_𝐿1(Ω),‖𝑔̃^𝑛‖_𝐿¹_(Ω)≤‖𝑔‖_𝐿¹_(Ω),(2.4) and let {𝑔̃^𝑛}_𝑛∈ℕ, {𝑢^𝑛0̃}_𝑛∈ℕ be another smooth approximation of the data with ‖‖𝑢^𝑛0̃‖‖_𝐿1(Ω)≤‖‖𝑢₀‖‖_𝐿1(Ω),‖𝑔̃^𝑛‖_𝐿¹_(Ω)≤‖𝑔‖_𝐿¹_(Ω).(2.5) Assume that 𝑢,𝑢̃ are two approximated solutions to problem (1.1), obtained as limit of the solutions to the following two approximated problems, respectively, 𝑢^𝑛𝑡−div(𝑎(𝑥)∇𝑢𝑢^𝑛)=+0𝜆,𝑓(o𝑢n^𝑛)𝜕=Ω𝑔×^𝑛+ℝ₊div𝐺,inΩ×ℝ⁺,𝑢(𝑥,0)=𝑢_𝑛0,𝑢_𝑛𝑡̃,inΩ,(2.6)−div(𝑎(𝑥)∇𝑢𝑢^𝑛̃)=+0𝜆,𝑓(o𝑢n^𝑛̃)𝜕=Ω𝑔×̃^𝑛+ℝ₊div𝐺,inΩ×ℝ⁺,𝑢(𝑥,0)=𝑢_𝑛0̃,,inΩ.(2.7) Now we prove that 𝑢=𝑢̃. For any 𝑘>0, define 𝜓_𝑘(𝑠) as 𝜓_𝑘(𝑠)=⎧⎪⎨⎪⎩𝑘𝑠−,,𝑘,𝑠𝑠|𝑠|><≤𝑘−𝑘𝑘.(2.8) Let Ψ_𝑘(𝜎)=∫^𝜎0𝜓_𝑘(𝑠)𝑑𝑠 be its primitive function. Taking 𝜓_𝑘(𝑢^𝑛−𝑢^𝑛̃) as a test function in (2.6) and (2.7), we deduce that _ΩΨ=_𝑘(𝜆𝑢^𝑛−_𝑄𝑇𝑢^𝑛̃)(𝑡)𝑑𝑥−_ΩΨ_𝑘𝑢^𝑛0−𝑢^𝑛0̃𝑑𝑥+_𝑄𝑇𝑎(𝑥)∇𝜓_𝑘(𝑢^𝑛−𝑢^𝑛̃)²𝑑𝑥𝑑𝑡≤(𝑓(𝑢^𝑛̃)−𝑓(𝑢^𝑛))𝜓_𝑘(𝑢^𝑛−𝑢^𝑛̃)𝑑𝑥𝑑𝑡+_𝑄𝑇(𝑔^𝑛−𝑔̃^𝑛)𝜓_𝑘(𝑢^𝑛−𝑢^𝑛̃)𝑑𝑥𝑑𝑡_𝑄𝑇𝜆𝑙(𝑢^𝑛−𝑢^𝑛̃)𝜓_𝑘(𝑢^𝑛−𝑢^𝑛̃)𝑑𝑥𝑑𝑡+_𝑄𝑇||𝑔^𝑛−𝑔̃^𝑛||𝑑𝑥𝑑𝑡.(2.9) Hence, from the assumptions on 𝑓, we get _ΩΨ_𝑘(𝑢^𝑛−𝑢^𝑛̃)(𝑡)𝑑𝑥≤2𝑙𝜆^𝑇0_ΩΨ_𝑘(𝑢^𝑛−𝑢^𝑛̃)𝑑𝑥𝑑𝑡+_𝑄𝑇||𝑔^𝑛−𝑔̃^𝑛||𝑑𝑥𝑑𝑡+_ΩΨ_𝑘𝑢^𝑛0−𝑢^𝑛0̃𝑑𝑥.(2.10) Let 𝑛→∞, we have _ΩΨ_𝑘(𝑢−𝑢)̃(𝑡)𝑑𝑥≤2𝑙𝜆^𝑇0_ΩΨ_𝑘(𝑢−𝑢)̃(𝑡)𝑑𝑥𝑑𝑡.(2.11) Thus for all 𝑘>0, we have sup_[]0,𝑇_ΩΨ_𝑘(𝑢̃−𝑢)(𝑡)𝑑𝑥≤2𝑙𝑇sup_[]0,𝑇_ΩΨ_𝑘(𝑢̃−𝑢)𝑑𝑥.(2.12) Taking 𝑇′ small enough such that 2𝑙𝜆𝑇′<1, we deduce that Ψ_𝑘(𝑢−𝑢)=0 for all 𝑘>0 in 𝑄_𝑇, thus 𝑢≡𝑢 in 𝑄_𝑇. 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 Theorem 1.1, there exists at least one approximated solution 𝑣 to the stationary problem of corresponding to problem (1.1), with (i)if 𝑝≥(2𝑁−2)/(𝑁−2), we have 𝑣∈𝑊^01,𝑞(Ω) for 𝑞<(2𝑝−2)/𝑝;(ii)if 2≤𝑝<(2𝑁−2)/(𝑁−2), we have 𝑣∈𝑊^01,𝑞(Ω) for 𝑞<𝑁/(𝑁−1).

Remark 2.4. Note that if 𝑣 is an approximated solution to following problem −div(𝑎(𝑥)∇𝑣𝑣)+=𝜆0𝑓,(𝑣o)n=𝜕𝑔Ω+,div𝐺,inΩ,(2.13) then there is a sequence {𝑣^𝑛} converges to 𝑣, where 𝑣^𝑛 is the solution of the corresponding approximated problem −div(𝑎(𝑥)∇𝑣^𝑛𝑣)+𝜆𝑓(𝑣^𝑛)=𝑔^𝑛+div𝐺,inΩ,_𝑛=0,on𝜕Ω.(2.14) And hence 𝑣^𝑛 is a solution of parabolic equations 𝑣^𝑛𝑡−div(𝑎(𝑥)∇𝑣𝑣^𝑛_𝑛)+𝜆𝑓(𝑣^𝑛)=𝑔^𝑛+div𝐺,inΩ×ℝ⁺,𝑣_𝑛=0,on𝜕Ω×ℝ⁺,(0)=𝑣^𝑛(𝑥),inΩ.(2.15) Thus, 𝑣 is an approximated solution of problem (1.1) with initial data 𝑢₀=𝑣(𝑥).

Under the assumptions of Theorem 1.2, the problem turns out to be 𝑢_𝑡−div𝑢(=𝑎(0𝑥,)∇o𝑢n)=𝜕Ω𝜇,×ℝin₊Ω×ℝ⁺,𝑢(𝑥,0)=𝑢₀,,inΩ.(2.16) 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 Theorem 1.2, 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 {𝜇^𝑛}_𝑛∈ℕ, {𝑢^𝑛0}_𝑛∈ℕ of data 𝜇 and 𝑢₀, such that the solution of the approximated problem of (2.16) with data 𝜇^𝑛 and 𝑢^𝑛0 converges to 𝑢. Let ℎ∈𝐶^∞𝑐(𝑄_𝑇) and 𝜔 be the solution of the the following parabolic problem −𝜔_𝑡−div𝜔=𝜔𝑎^∗(0(𝑥,𝑥,)𝑇∇o)n𝜔=𝜕0Ω=,×ℎi,n(0Ωi,n𝑇,Ω),×(0,𝑇),(2.17) where 𝑎^∗(𝑥) is the transposed matrix of 𝑎(𝑥). Taking 𝜔 as a test function in the approximated problem and taking 𝑢^𝑛 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]). Let 𝑋, 𝑌 be two Banach spaces, let 𝑋 be separable, reflexive, and let 𝑋⊂𝑌 with dual 𝑋^∗. Suppose that {𝑢^𝑛} is uniformly bounded in 𝐿^∞((0,𝑇);𝑋) with esssup_{𝑡∈[0,𝑇]}‖𝑢^𝑛(𝑡)‖_𝑋≤𝐶,(2.18) and that 𝑢^𝑛→𝑢 weakly in 𝐿^𝑟((0,𝑇);𝑋) for some 𝑟∈(1,∞). Then esssup_{𝑡∈[0,𝑇]}‖𝑢(𝑡)‖_𝑋≤𝐶.(2.19) Moreover, if 𝑢∈𝐶([0,𝑇];𝑌), then in fact sup_{𝑡∈[0,𝑇]}‖𝑢(𝑡)‖_𝑋≤𝐶.(2.20)

Lemma 2.7 (see [16]). Let 𝑋,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 𝑆(𝑡)𝐵⊂𝐵₀,∀𝑡≥𝑇.(2.21) 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 solution 𝑢 to problem (1.1).

Theorem 3.1. Under the assumptions of Theorem 2.2, 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 𝑤_𝑡−div(𝑎(𝑥)∇𝑤)𝑤+=𝜆𝑓0,(𝑣o+n𝑤𝜕)Ω−×𝜆𝑓ℝ₊(𝑣)=0,inΩ×ℝ⁺,𝑤(𝑥,0)=𝑢₀,−𝑣,inΩ.(3.1) Moreover, we have (i)𝑤∈𝐿^∞((𝛿,𝑇);𝐿^𝑞(Ω)) for any 0<𝛿<𝑇,1≤𝑞<∞. Moreover, there exists a constant 𝑀_𝑞 and a time 𝑡_𝑞(𝑢₀,𝑔,𝐺) such that ‖𝑤(𝑡)‖_𝐿^𝑞_(Ω)≤𝑀_𝑞forall𝑡≥𝑡_𝑞(𝑢₀,𝑔,𝐺).(ii)𝑤∈𝐿^∞((𝛿,𝑇);𝐻¹⁰(Ω)) for any 0<𝛿<𝑇. Moreover, there exists a constant 𝜌 and a time 𝑇₀(𝑢₀,𝑔,𝐺) such that ‖𝑤(𝑡)‖_𝐻¹⁰_(Ω)≤𝜌forall𝑡≥𝑇₀(𝑢₀,𝑔,𝐺).

Proof. We follow the lines of [15, 28]. Let {𝑔^𝑛} be a sequence of smooth data which converges to 𝑔 in 𝐿¹(Ω) and ‖𝑔^𝑛‖_𝐿¹_(Ω)≤‖𝑔‖_𝐿¹_(Ω). Let 𝑣^𝑛 be a solution of the following approximated problem for each 𝑛, −div(𝑎(𝑥)∇𝑣^𝑛)𝑣+𝜆𝑓(𝑣^𝑛)=𝑔^𝑛+div𝐺,inΩ,_𝑛=0,on𝜕Ω.(3.2) Then 𝑣^𝑛 converges (up to subsequences) to an approximated solution 𝑣 strongly in 𝐿¹(Ω), and weakly in 𝑊^01,𝑞(Ω),1≤𝑞<𝑁/(𝑁−1). Let {𝑢^𝑛} be a sequence of solutions to the following approximated problem 𝑢^𝑛𝑡−div(𝑎(𝑥)∇𝑢𝑢^𝑛_𝑛)+𝜆𝑓(𝑢^𝑛)=𝑔^𝑛(𝑥)+div𝐺,inΩ×ℝ⁺,𝑢(𝑥_𝑛)=0,on𝜕Ω×ℝ⁺,(𝑥,0)=𝑢^𝑛0,inΩ,(3.3) where 𝑢^𝑛0 converges to 𝑢₀ with ‖𝑢^𝑛0‖_𝐿¹_(Ω)≤‖𝑢₀‖_𝐿¹_(Ω). Similar to [8, 29], we know that 𝑢^𝑛→𝑢weaklyin𝐿^𝑞(0,𝑇);𝑊^01,𝑞(Ω),𝑞<𝑁+2𝑢𝑁+1_𝑛,⟶𝑢in𝐶[0,𝑇];𝐿¹(Ω).(3.4) Now let 𝑤^𝑛(𝑡)=𝑢^𝑛(𝑡)−𝑣^𝑛. Then 𝑤^𝑛 satisfies 𝑤^𝑛𝑡−div(𝑎(𝑥)∇𝑤^𝑛)𝑤+_𝑛𝜆𝑓(𝑣^𝑛+𝑤^𝑛)−𝜆𝑓(𝑣^𝑛)=0,inΩ×ℝ⁺,𝑤_𝑛=0,on𝜕Ω×ℝ⁺,(𝑥,0)=𝑢^𝑛0−𝑣^𝑛,inΩ.(3.5) Similarly, we have 𝑤^𝑛 (up to subsequences) converges to the approximated solution 𝑤 of problem (3.1) in 𝐶([0,𝑇];𝐿¹(Ω)) and weakly in 𝐿^𝑞((0,𝑇);𝑊^01,𝑞(Ω)),𝑞<(𝑁+2)/(𝑁+1).
Now we prove (i). Taking 𝜓₁(𝑢^𝑛) as test function in (3.3) (for simplicity we take 𝜆=1), we deduce that 𝑑𝑑𝑡_ΩΨ₁(𝑢^𝑛)𝑑𝑥+_Ω𝑓(𝑢^𝑛)𝜓₁(𝑢^𝑛)𝑑𝑥≤‖𝑔‖_𝐿¹_(Ω)+𝐶‖𝐺‖_𝐿²_(Ω).(3.6) Since 𝑓(𝑢^𝑛)𝜓₁(𝑢^𝑛)≥𝐶||𝑢^𝑛||^𝑝−1−𝐶||𝜓₁(𝑢^𝑛)||,(3.7) we have 𝑑𝑑𝑡_ΩΨ₁(𝑢^𝑛)𝑑𝑥+𝐶_Ω||Ψ₁(𝑢^𝑛)||𝑑𝑥≤‖𝑔‖_𝐿¹_(Ω)+𝐶‖𝐺‖_𝐿²_(Ω)+𝐶||Ω||.(3.8) The Gronwall's inequality implies that _ΩΨ₁(𝑢^𝑛(𝑡))𝑑𝑥≤‖‖𝑢₀‖‖_𝐿1(Ω)𝑒^−𝐶𝑡+𝐶||Ω||+𝐶‖𝑔‖_𝐿¹_(Ω)+𝐶‖𝐺‖_𝐿²_(Ω).(3.9) Noticing that _Ω||𝑢^𝑛(𝑡)||𝑑𝑥≤_ΩΨ₁(𝑢^𝑛(𝑡))𝑑𝑥+||Ω||,(3.10) we obtain that _Ω||𝑢^𝑛(𝑡)||𝑑𝑥≤‖‖𝑢₀‖‖_𝐿1(Ω)𝑒^−𝐶𝑡+𝐶||Ω||+𝐶‖𝑔‖_𝐿¹_(Ω)+𝐶‖𝐺‖_𝐿²_(Ω),𝑡≥0.(3.11) Moreover, integrating (3.6) between 𝑡 and 𝑡+1 and using (3.7) we have ^𝑡𝑡+1_Ω||𝑢^𝑛||^𝑝−1𝑑𝑥𝑑𝜉≤𝐶‖‖𝑢₀‖‖_𝐿1(Ω)𝑒^−𝐶𝑡+||Ω||+‖𝑔‖_𝐿¹_(Ω)+‖𝐺‖_𝐿²_(Ω).(3.12) Similarly, taking 𝜓₁(𝑣^𝑛) as test function in (3.2), we can deduce that _Ω||𝑓(𝑣^𝑛)||𝑑𝑥≤𝐶_Ω||𝑣^𝑛||^𝑝−1+1𝑑𝑥≤𝐶‖𝑔‖_𝐿¹_(Ω)+‖𝐺‖_𝐿²_(Ω)+||Ω||.(3.13) Hence, ^𝑡𝑡+1_Ω||𝑤^𝑛||^𝑝−1𝑑𝑥𝑑𝜉≤𝐶‖𝑔‖_𝐿¹_(Ω)+‖𝐺‖_𝐿²_(Ω)+‖‖𝑢₀‖‖_𝐿1(Ω)+||Ω||,(3.14) with 𝐶 independent of 𝑛, for 𝑡≥0.
Now we use bootstrap method in the case 𝑝≥3. The case 2≤𝑝<3 can be treated similarly with minor modifications. Multiplying (3.5) by |𝑤^𝑛|^𝑞0−2𝑤^𝑛,𝑞₀=𝑝−1≥2, and integrating on Ω, we obtain 1𝑞₀𝑑𝑑𝑡_Ω||𝑤^𝑛||^𝑞0𝑑𝑥+𝑞₀−1𝑐_Ω||∇𝑤^𝑛||²||𝑤^𝑛||^𝑞0−2𝑑𝑥≤𝑙_Ω||𝑤^𝑛||^𝑞0𝑑𝑥.(3.15) Since |∇𝑤^𝑛|²|𝑤^𝑛|^𝑞0−2=(2/𝑞₀)²|∇(|𝑤^𝑛|^{(𝑞0−2)/2}𝑤^𝑛)|², we deduce that 𝑑𝑑𝑡_Ω||𝑤^𝑛||^𝑞0𝑑𝑥+_Ω|||∇||𝑤_𝑛||^{(𝑞0−2)/2}𝑤^𝑛|||²𝑑𝑥≤𝐶_Ω||𝑤^𝑛||^𝑞0𝑑𝑥.(3.16) Integrating (3.16) between 𝑠 and 𝑡+1(𝑡≤𝑠<𝑡+1), it yields _Ω||𝑤^𝑛(𝑡+1)||^𝑞0𝑑𝑥≤𝐶^𝑠𝑡+1_Ω||𝑤^𝑛||^𝑞0𝑑𝑥𝑑𝜉+_Ω||𝑤^𝑛(𝑠)||^𝑞0𝑑𝑥.(3.17) Integrating the above inequality with respect to 𝑠 between 𝑡 and 𝑡+1, we get _Ω||𝑤^𝑛(𝑡+1)||^𝑞0𝑑𝑥≤𝐶^𝑡𝑡+1_Ω||𝑤^𝑛||^𝑞0𝑑𝑥𝑑𝜉.(3.18) Therefore, _Ω||𝑤^𝑛(𝑡)||^𝑞0𝑑𝑥≤𝐶,∀𝑡≥1.(3.19) Integrating (3.16) on [𝑡,𝑡+1] for 𝑡≥1, we deduce that ^𝑡𝑡+1_Ω|∇||𝑤^𝑛||^{(𝑞0−2)/2}𝑤^𝑛|²𝑑𝑥𝑑𝜉≤𝐶^𝑡𝑡+1_Ω||𝑤^𝑛(𝜉)||^𝑞0𝑑𝑥𝑑𝜉+𝐶_Ω||𝑤^𝑛(𝑡)||^𝑞0𝑑𝑥≤𝐶.(3.20) Note that (3.20) insures that, for any 𝑡≥1, there exists at least a 𝑡₀∈[𝑡,𝑡+1] such that _Ω|||∇||𝑤_𝑛𝑡₀||^{(𝑞0−2)/2}𝑤^𝑛𝑡₀|||²𝑑𝑥≤𝐶.(3.21) Standard Sobolev imbedding implies that _Ω||𝑤^𝑛𝑡₀||^{𝑞0(𝑁/(𝑁−2))}𝑑𝑥≤𝐶.(3.22) Now multiplying (3.5) by |𝑤^𝑛|^𝑞1−2𝑤^𝑛,𝑞₁=(𝑁/(𝑁−2))𝑞₀, we have 𝑁−2𝑁𝑞₀𝑑𝑑𝑡_Ω||𝑤^𝑛||^{(𝑁/(𝑁−2))𝑞0}𝑑𝑥+𝐶𝑐,𝑞₀,𝑁_Ω|||∇||𝑤_𝑛||^{(𝑁𝑞0−2𝑁+4)/(2𝑁−4)}𝑤^𝑛≤𝑙_Ω|||²𝑑𝑥||𝑤^𝑛||^{(𝑁/(𝑁−2))𝑞0}𝑑𝑥.(3.23) Using Hölder inequality, and Young inequality we deduce that _Ω|𝑤^𝑛|^{(𝑁/(𝑁−2))𝑞0}𝑑𝑥≤𝜀^_Ω|||∇||𝑤_𝑛||^{(𝑁𝑞0−2𝑁+4)/(2𝑁−4)}𝑤^𝑛|||²𝑑𝑥+𝐶_𝜀_Ω||𝑤^𝑛||^𝑞0𝑑𝑥^{𝑁/(𝑁−2)}.(3.24) Taking (3.24) into (3.23), it yields 𝑑𝑑𝑡_Ω||𝑤^𝑛||^{(𝑁/(𝑁−2))𝑞0}𝑑x+𝐶^𝑐,𝑞₀,𝑁_Ω|||∇||𝑤_𝑛||^{(𝑁𝑞0−2𝑁+4)/(2𝑁−4)}𝑤^𝑛≤𝐶_𝜀|||²𝑑𝑥_Ω||𝑤^𝑛||^𝑞0𝑑𝑥^{𝑁/(𝑁−2)}.(3.25) Integrating (3.25) between 𝑡₀ and 𝑡₀+𝑠,0<𝑠≤1, we have _Ω||𝑤^𝑛𝑡₀+𝑠||^{(𝑁/(𝑁−2))𝑞0}𝑑𝑥≤_Ω||𝑤^𝑛𝑡₀||^{(𝑁/(𝑁−2))𝑞0}𝑑𝑥+𝐶^𝜀_Ω||𝑤^𝑛||^𝑞0𝑑𝑥^{𝑁/(𝑁−2)}.(3.26) Therefore, from (3.19) and (3.22) we get _Ω||𝑤^𝑛(𝑡)||^{(𝑁/(𝑁−2))𝑞0}𝑑𝑥=_Ω||𝑤^𝑛(𝑡)||^𝑞1𝑑𝑥≤𝐶,∀𝑡≥2,(3.27) with 𝐶 independent of 𝑛. Integrating (3.25) between 𝑡 and 𝑡+1 for 𝑡≥2, we obtain ^𝑡𝑡+1_Ω|||∇||𝑤_𝑛||^{(𝑞1−2)/2}𝑤^𝑛|||²𝑑𝑥𝑑𝜉=^𝑡𝑡+1_Ω|∇(||𝑤^𝑛||^{(𝑁𝑞0−2𝑁+4)/(2𝑁−4)}𝑤^𝑛)|²𝑑𝑥𝑑𝜉≤𝐶.(3.28) Similar to (3.22), for any 𝑡≥2, there exists at least a 𝑡₀∈[𝑡,𝑡+1] such that _Ω|𝑤^𝑛(𝑡₀)|^{𝑞0(𝑁2/(𝑁−2)2)}𝑑𝑥≤𝐶.(3.29) Bootstrap the above processes, we can deduce that _Ω||𝑤^𝑛(𝑡)||^𝑞𝑘𝑑𝑥≤𝐶,∀𝑡≥𝑇_𝑘,(3.30) with 𝑞_𝑘=(𝑁/(𝑁−2))^𝑘𝑞₀, and 𝐶 independent of 𝑛. Note that 𝑤^𝑛→𝑤 in 𝐶([0,𝑇];𝐿¹(Ω)) and 𝑤∈𝐶([0,𝑇];𝐿¹(Ω)). From Lemma 2.6, we have ‖𝑤(𝑡)‖^𝑞𝑘_{𝐿𝑞𝑘(Ω)}=_Ω||𝑤(𝑡)||^𝑞𝑘𝑑𝑥≤𝐶,∀𝑡≥𝑇_𝑘.(3.31) 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 𝑤^𝑛 and deduce that 12𝑑𝑑𝑡_Ω||𝑤^𝑛||²𝑑𝑥+𝑐_Ω||∇𝑤^𝑛||²𝑑𝑥≤𝑙_Ω||𝑤^𝑛||²𝑑𝑥,(3.32) integrating over [𝑡,𝑡+1],𝑡≥𝑇′, we get ^𝑡𝑡+1_Ω||∇𝑤^𝑛||²𝑑𝑥𝑑𝑡≤𝐶,(3.33) with 𝐶 independent of 𝑛. Now, multiplying (3.5) with 𝑤^𝑛𝑡, we obtain _Ω||𝑤^𝑛𝑡||²𝑑𝑥+𝑑𝑑𝑡_Ω𝑎(𝑥)||∇𝑤^𝑛||²𝑑𝑥+𝑑𝑑𝑡_Ω(𝐹(𝑤^𝑛+𝑣^𝑛)−𝐹(𝑣^𝑛)−𝑓(𝑣^𝑛)𝑤^𝑛)𝑑𝑥=0,(3.34) where 𝐹(𝑣+𝜎)=∫^𝜎0𝑓(𝑣+𝑠)𝑑𝑠. Integrating (3.34) between 𝑠 and 𝑡+1(𝑡≤𝑠<𝑡+1) gives _Ω||∇≤𝑤𝐶^𝑛(𝑡_Ω+1)||²𝑑𝑥+_Ω(𝐹(𝑤^𝑛(𝑡+1)+𝑣^𝑛)−𝐹(𝑣^𝑛)−𝑓(𝑣^𝑛)𝑤^𝑛(𝑡+1))𝑑𝑥||∇𝑤^𝑛(𝑠)||²𝑑𝑥+𝐶_Ω(𝐹(𝑤^𝑛(𝑠)+𝑣^𝑛)−𝐹(𝑣^𝑛)−𝑓(𝑣^𝑛)𝑤^𝑛(𝑠))𝑑𝑥.(3.35) Now, integrating the above inequality with respect to 𝑠 between 𝑡 and 𝑡+1 we have _Ω||∇≤𝑤𝐶^𝑛(𝑡+_𝑡𝑡+11)||²𝑑𝑥+_Ω(𝐹(𝑤^𝑛(𝑡+1)+𝑣^𝑛)−𝐹(𝑣^𝑛)−𝑓(𝑣^𝑛)𝑤^𝑛(𝑡+1))𝑑𝑥_Ω||∇𝑤^𝑛||²𝑑𝑥𝑑𝜉+𝐶^𝑡𝑡+1||||_Ω(𝐹(𝑤^𝑛+𝑣^𝑛)−𝐹(𝑣^𝑛)−𝑓(𝑣^𝑛)𝑤^𝑛)𝑑𝑥||||𝑑𝜉.(3.36) Since _Ω(𝐹(𝑤^𝑛+𝑣^𝑛)−𝐹(𝑣^𝑛)−𝑓(𝑣^𝑛)𝑤^𝑛)𝑑𝑥=_Ω¹⁰𝑓(𝑣_𝑛(𝑓(𝑣^𝑛+𝑠𝑤^𝑛)−𝑓(𝑣^𝑛))𝑤^𝑛𝑑𝑠𝑑𝑥,+𝜏𝑤^𝑛)−𝑓(𝑣^𝑛)=¹⁰𝑓^(𝑣^𝑛+𝜃𝜏𝑤^𝑛)𝜏𝑤^𝑛𝑑𝜃,0≤𝜏≤1.(3.37) We deduce that ||||_Ω(𝐹(𝑤^𝑛+𝑣^𝑛)−𝐹(𝑣^𝑛)−𝑓(𝑣^𝑛)𝑤^𝑛)𝑑𝑥||||≤_Ω||𝑓^𝑣^𝑛+𝜏^𝑤^𝑛||||𝑤^𝑛||²𝑑𝑥,0≤𝜏^≤1.(3.38) From the assumption (1.4) on 𝑓, we have ||||_Ω(≤𝐹(𝑤^𝑛+𝑣^𝑛)−𝐹(𝑣^𝑛)−𝑓(𝑣^𝑛)𝑤^𝑛)𝑑𝑥_Ω||||≤𝐶𝐶‖𝑣_𝑛||𝑣^𝑛||^𝑝−2+||𝑤^𝑛||^𝑝−2+1||𝑤^𝑛||²𝑑𝑥‖^𝐿𝑝−2_𝑝−1(Ω)+‖𝑤^𝑛‖^2𝐿_{2𝑝−2(Ω)}+‖𝑤^𝑛‖^𝑝𝐿_𝑝(Ω)+1.(3.39) Using results in (3.13) and (3.30), we know that ||||_Ω(𝐹(𝑤^𝑛+𝑣^𝑛)−𝐹(𝑣^𝑛)−𝑓(𝑣^𝑛)𝑤^𝑛)𝑑𝑥||||≤𝐶,∀𝑡≥𝑇^.(3.40) Set 𝑇₀=max{𝑇^,𝑇^}. Combining (3.33), (3.34), (3.36), and (3.40) we have _Ω||∇𝑤^𝑛(𝑡)||²𝑑𝑥≤𝐶,∀𝑡≥𝑇₀+1,(3.41)_𝑡𝑡+1_Ω||𝑤^𝑛𝑡||²𝑑𝑥𝑑𝜉≤𝐶,∀𝑡≥𝑇₀+1.(3.42) From Lemma 2.6, we obtain ‖𝑤‖_𝐻¹⁰_(Ω)≤𝐶,∀𝑡≥𝑇₀+1.(3.43) 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 𝑆₁(𝑡)𝑢₀−𝑣(𝑥)=𝑆(𝑡)𝑢₀−𝑣(𝑥).(4.1) 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 Theorem 1.1, the semigroup {𝑆₁(𝑡)}_𝑡≥0 possesses a global attractor 𝒜_𝑣 in 𝐿^𝑟(Ω)∩𝐻¹⁰(Ω),1≤𝑟<∞, that is, 𝒜_𝑣 is compact, invariant in 𝐿^𝑟(Ω)∩𝐻¹⁰(Ω),1≤𝑟<∞, and attracts every bounded initial set of 𝐿¹(Ω) in the norm topology of 𝐿^𝑟(Ω)∩𝐻¹⁰(Ω),1≤𝑟<∞.

Proof. From Theorem 3.1, 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 𝐿^𝑟(Ω)∩𝐻¹⁰(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 _Ω||𝑤^𝑛𝑡||²𝑑𝑥≤𝐶,for𝑡largeenough.(4.2) Actually, differentiating (3.5) in time and denoting 𝑤^𝑛=𝑤^𝑛𝑡, we have 𝑤^𝑛𝑡−div𝑎(𝑥)∇𝑤^𝑛+𝑓′(𝑣^𝑛+𝑤^𝑛)𝑤^𝑛=0.(4.3) Multiplying (4.3) by 𝑤^𝑛 and using (1.3), we deduce that 𝑑𝑑𝑡_Ω||𝑤^𝑛||²𝑑𝑥≤𝑙_Ω||𝑤^𝑛||²𝑑𝑥.(4.4) Integrating the above inequality between 𝑠 and 𝑡+1(𝑡≤𝑠<𝑡+1) gives _Ω||𝑤^𝑛(𝑡+1)||²𝑑𝑥≤2𝑙^𝑠𝑡+1_Ω||𝑤^𝑛||²𝑑𝑥𝑑𝜉+_Ω||𝑤^𝑛(𝑠)||²𝑑𝑥.(4.5) Integrating the above inequality with respect to 𝑠 between 𝑡 and 𝑡+1, using (3.42) we obtain _Ω||𝑤^𝑛(𝑡)||²𝑑𝑥≤(2𝑙+1)^𝑡𝑡+1_Ω||𝑤^𝑛||²𝑑𝑥𝑑𝜉≤(2𝑙+1)𝐶,∀𝑡≥𝑇₀+1.(4.6) Now we prove the asymptotic compactness of the semigroup {𝑆₁(𝑡)}_𝑡≥0, that is, for any sequences {𝑤_0,𝑘}⊂𝐵₁,𝑡_𝑘→∞, sequence {𝑤_𝑘(𝑡_𝑘)} has convergent subsequences, where 𝑤_𝑘(𝑡)=𝑆₁(𝑡)𝑤_0,𝑘. Since {𝑆₁(𝑡)}_𝑡≥0 is compact in 𝐿^𝑟(Ω),1≤𝑟<∞, there is a subsequence {𝑤_𝑘𝑖(𝑡_𝑘𝑖)}, which is a Cauchy sequence in 𝐿^𝑟(Ω),1≤𝑟<∞. Denoting 𝑢_𝑘𝑖(𝑡_𝑘𝑖)=𝑤_𝑘𝑖(𝑡_𝑘𝑖)+𝑣,𝑢_𝑘𝑗(𝑡_𝑘𝑗)=𝑤_𝑘𝑗(𝑡_𝑘𝑗)+𝑣, we deduce that 𝑐‖‖𝑤_𝑘𝑖(𝑡_𝑘𝑖)−𝑤_𝑘𝑗𝑡^𝑘_𝑗‖‖^2𝐻_10(Ω)=≤𝜕div𝑎(𝑥)∇𝑤_𝑘𝑖𝑡_𝑘𝑖+𝑣−div𝑎(𝑥)∇𝑤_𝑘𝑗𝑡^𝑘_𝑗+𝑣,𝑤^𝑘_𝑖𝑡_𝑘𝑖−𝑤_𝑘𝑗𝑡^𝑘_𝑗_𝑡≤‖‖𝜕𝑤_𝑘𝑖𝑡_𝑘𝑖−𝜕_𝑡𝑤_𝑘𝑗𝑡^𝑘_𝑗+𝑓𝑢^𝑘_𝑖𝑡_𝑘𝑖−𝑓𝑢_𝑘𝑗𝑡^𝑘_𝑗,𝑤^𝑘_𝑖𝑡_𝑘𝑖−𝑤_𝑘𝑗𝑡^𝑘_𝑗_𝑡𝑤_𝑘𝑖(𝑡_𝑘𝑖)−𝜕_𝑡𝑤_𝑘𝑗(𝑡_𝑘𝑗)‖‖_𝐿2(Ω)‖‖𝑤_𝑘𝑖(𝑡_𝑘𝑖)−𝑤_𝑘𝑗𝑡^𝑘_𝑗‖‖_𝐿2(Ω)+‖‖𝑓𝑤_𝑘𝑖𝑡_𝑘𝑖+𝑣−𝑓𝑤_𝑘𝑗𝑡^𝑘_𝑗+𝑣‖‖_𝐿𝜎(Ω)‖‖𝑤_𝑘𝑖𝑡_𝑘𝑖−𝑤_𝑘𝑗𝑡^𝑘_𝑗‖‖_{𝐿𝜎′(Ω)}.(4.7) We then conclude form (4.6) that {𝑤_𝑘𝑖(𝑡_𝑘𝑖)} is a Cauchy sequence in 𝐻¹⁰(Ω), and thus {𝑆₁(𝑡)}_𝑡≥0 is asymptotically compact in 𝐿^𝑟(Ω)∩𝐻¹⁰(Ω),1≤𝑟<∞.
Using Lemma 2.7, we conclude that {𝑆₁(𝑡)}_𝑡≥0 possesses a global attractor 𝒜_𝑣, which is compact, invariant in 𝐿^𝑟(Ω)∩𝐻¹⁰(Ω), and attracts every bounded initial sets of 𝐿¹(Ω) in the topology of 𝐿^𝑟(Ω)∩𝐻¹⁰(Ω).

Completion of the Proof of Theorem 1.1
Note that 𝒜_𝑣=∩_𝑠≥0∪_𝑡≥𝑠𝑆₁(𝑡)𝐵_2𝐿^1(Ω)=𝒜∩=∩_𝑠≥0∪_{𝑡≥𝑠𝑠≥0}𝑆₁(𝑡)𝐵_1𝐿^1(Ω)=∩_𝑠≥0∪_𝑡≥𝑠𝑆(𝑡)𝐵₀−𝑣^𝐿1(Ω),∪_𝑡≥𝑠𝑆(𝑡)𝐵_0𝐿^1(Ω).(4.8) Thus we have 𝒜=𝒜_𝑣+𝑣.(4.9) The above relation between 𝒜 and 𝒜_𝑣 implies the conclusion of Theorem 1.1 directly.

Proof of Theorem 1.2. Let 𝑢(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 𝐿^𝑟(Ω),1≤𝑟<∞, as 𝑡→∞. Moreover from the asymptotic compactness of the semigroup 𝑆₁(𝑡), we know that 𝑤(𝑡) converges to 0 in 𝐻¹⁰(Ω) as 𝑡→∞. Else, we have a sequence 𝑡_𝑛→∞, such that 𝐶>‖𝑤(𝑡_𝑛)‖_𝐻¹⁰_(Ω)≥𝜖>0. Since the semigroup 𝑆₁(𝑡) is asymptotically compact, there is a subsequence 𝑡_𝑛𝑗→∞, such that 𝑤(𝑡_𝑛𝑗) converges to a function 𝜒 in 𝐻¹⁰(Ω) and hence in 𝐿¹(Ω). Thus 𝜒=0. A contradiction!

Acknowledgments

The authors want to thank the referee for the careful reading of the manuscript. This work is partially supported by the 211 Project of Anhui University (KJTD002B/32030015) and the NSFC Grant (11071001, 11271019).