Asymptotic Behavior of Approximated Solutions to Parabolic Equations with Irregular Data

and Applied Analysis 3 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 Ω ∩H1 0 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 Lp−1 Ω ∩W 0 Ω with q < max{N/ N − 1 , 2p − 2 /p}, and attracts every bounded subset of L1 Ω in the norm topology of L Ω ∩H1 0 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 Ω ∩H1 0 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 Ω ∩H1 0 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 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 function u is called an approximated solution of problem 1.1 , if u ∈ L1 0, T ;W 0 Ω , f u ∈ L1 QT for any T > 0, and − ∫


Introduction
We consider the asymptotic behavior of solutions to the following equations where Ω is a bounded domain in R N N ≥ 2 with smooth boundary ∂Ω, u 0 ∈ L 1 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 Ω such that f u ∈ L 1 Q T for any T > 0, and 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 L 1 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 L 1 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 1,q 0 Ω with q < max{N/ N −1 , 2p− 2 /p}, which attracts every bounded subset of L 1 Ω in the norm of L r Ω ∩ H 1 0 Ω , for any r ∈ 1, ∞ .

Abstract and Applied Analysis 3
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 ∩ H 1 0 Ω , for any r ∈ 1, ∞ , though they all lie in some less regular spaces.
Our main results can be stated as follows.
Theorem 1.2. Assume that u 0 ∈ L 1 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 ∩ H 1 0 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 ∩ H 1 0 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 L 1 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.

Existence Results and Useful Lemmas
We begin this section by providing some existence results on the approximated solutions. Abstract and Applied Analysis for any ϕ ∈ C ∞ c 0, T × Ω , and moreover, u is obtained as limit of solutions to the following approximated problem where {μ n }, {u n 0 } is a smooth approximation of data μ, u 0 .
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 Hence, we need only to consider the following problem

2.3
The proof of existence part of the theorem is similar to 9 . Besides, one can prove u ∈ C 0, T ; L 1 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 n 0 } n∈N be a smooth approximation of data g and u 0 with and let { g n } n∈N , { u n 0 } n∈N be another smooth approximation of the data with Abstract and Applied Analysis 5 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,

2.7
Now we prove that u u. For any k > 0, define ψ k s as

2.8
Let Ψ k σ σ 0 ψ k s ds be its primitive function. Taking ψ k u n − u n as a test function in 2.6 and 2.7 , we deduce that

2.9
Hence, from the assumptions on f, we get

2.10
Let n → ∞, we have

Abstract and Applied Analysis
Thus for all k > 0, we have 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.
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 n λf v n g n div G, in Ω, v n 0, on ∂Ω.

2.14
And hence v n is a solution of parabolic equations

2.15
Thus, v is an approximated solution of problem 1.1 with initial data u 0 v x .
Under the assumptions of Theorem 1.2, the problem turns out to be

2.16
The existence of approximated solutions to problem 1. 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 n 0 } n∈N of data μ and u 0 , such that the solution of the approximated problem of 2.16 with data μ n and u n 0 converges to u. Let h ∈ C ∞ c Q T and ω be the solution of the the following parabolic problem

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 . and that u n → u weakly in L r 0, T ; X for some r ∈ 1, ∞ . Then ess sup t∈ 0,T u t X ≤ C.

2.19
Moreover, if u ∈ C 0, T ; Y , then in fact

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 .

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 u 0 , g, G such that w t L q Ω ≤ M q for all t ≥ t q u 0 , g, G .
ii w ∈ L ∞ δ, T ; H 1 0 Ω for any 0 < δ < T. Moreover, there exists a constant ρ and a time T 0 u 0 , g, G such that w t H 1 0 Ω ≤ ρ for all t ≥ T 0 u 0 , g, G . Proof. We follow the lines of 15, 28 . Let {g n } be a sequence of smooth data which converges to g in L 1 Ω and g n L 1 Ω ≤ g L 1 Ω . Let v n be a solution of the following approximated problem for each n, − div a x ∇v n λf v n g n div G, in Ω, v n 0, on ∂Ω.

3.2
Then v n converges up to subsequences to an approximated solution v strongly in L 1 Ω , and weakly in W 1,q 0 Ω , 1 ≤ q < N/ N−1 . Let {u n } be a sequence of solutions to the following approximated problem u n t − div a x ∇u n λf u n g n x div G, in Ω × R , u n x 0, on ∂Ω × R , u n x, 0 u n 0 , in Ω,

3.3
Abstract and Applied Analysis 9 where u n 0 converges to u 0 with u n 0 L 1 Ω ≤ u 0 L 1 Ω . Similar to 8, 29 , we know that

3.4
Now let w n t u n t − v n . Then w n satisfies

3.5
Similarly, we have w n up to subsequences converges to the approximated solution w of problem 3.1 in C 0, T ; L 1 Ω and weakly in L q 0, T ; W 1,q 0 Ω , q < N 2 / N 1 . Now we prove i . Taking ψ 1 u n as test function in 3.3 for simplicity we take λ 1 , we deduce that Since f u n ψ 1 u n ≥ C|u n | p−1 − C ψ 1 u n , 3.7 we have The Gronwall's inequality implies that Noticing that Ω |u n t |dx ≤ Ω Ψ 1 u n t dx |Ω|, 3.10 we obtain that Ω |u n t |dx ≤ u 0 L 1 Ω e −Ct C|Ω| C g L 1 Ω C G L 2 Ω , t ≥ 0.

3.11
Moreover, integrating 3.6 between t and t 1 and using 3.7 we have Similarly, taking ψ 1 v n as test function in 3.2 , we can deduce that Hence, 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 | q 0 −2 w n , q 0 p − 1 ≥ 2, and integrating on Ω, we obtain Since |∇w n | 2 |w n | q 0 −2 2/q 0 2 |∇ |w n | q 0 −2 /2 w n | 2 , we deduce that Integrating 3.16 between s and t 1 t ≤ s < t 1 , it yields Integrating the above inequality with respect to s between t and t 1, we get Therefore, Ω |w n t | q 0 dx ≤ C, ∀t ≥ 1.

3.19
Abstract and Applied Analysis 11 Integrating 3.16 on t, t 1 for t ≥ 1, we deduce that Note that 3.20 insures that, for any t ≥ 1, there exists at least a t 0 ∈ t, t 1 such that Standard Sobolev imbedding implies that Now multiplying 3.5 by |w n | q 1 −2 w n , q 1 N/ N − 2 q 0 , we have

3.23
Using Hölder inequality, and Young inequality we deduce that

3.26
Therefore, from 3.19 and 3.22 we get with C independent of n. Integrating 3.25 between t and t 1 for t ≥ 2, we obtain Similar to 3.22 , for any t ≥ 2, there exists at least a t 0 ∈ t, t 1 such that Bootstrap the above processes, we can deduce that with q k N/ N − 2 k q 0 , and C independent of n. Note that w n → w in C 0, T ; L 1 Ω and w ∈ C 0, T ; L 1 Ω . From Lemma 2.6, we have Taking k large enough, we get the second part of i proved. If the integration are taken over t, t δ 0 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 integrating over t, t 1 , t ≥ T , we get with C independent of n. Now, multiplying 3.5 with w n t , we obtain where F v σ σ 0 f v s ds. Integrating 3.34 between s and t 1 t ≤ s < t 1 gives

3.35
Abstract and Applied Analysis 13 Now, integrating the above inequality with respect to s between t and t 1 we have

3.37
We deduce that From the assumption 1.4 on f, we have

3.39
Using results in 3.13 and 3.30 , we know that Set T 0 max{T , T }. Thus we get the second part of ii proved. Taking integration over t, t δ 0 instead of t, t 1 , the first part of ii follows. The proof is completed now.

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 Then it is easy to verify that {S 1 t } t≥0 is a continuous semigroup in L 1 Ω − v and hence in L 1 Ω . From the results in Section 3, we know that the semigroup {S t } t≥0 possesses a global attractor A in L 1 Ω . To verify the second part of Theorem 1.1, we prove the following theorem.
and attracts every bounded initial set of L 1 Ω in the norm topology of L r Ω ∩ H 1 0 Ω , 1 ≤ r < ∞.
In the next, we prove the asymptotic compactness of {S 1 t } t≥0 . Before that we establish the following estimate Ω w n t 2 dx ≤ C, for t large enough.

4.2
Actually, differentiating 3.5 in time and denoting w n w n t , we have ≤ div a x ∇ w k i t k i v − div a x ∇ w k j t k j v , w k i t k i − w k j t k j ∂ t w k i t k i − ∂ t w k j t k j f u k i t k i − f u k j t k j , w k i t k i − w k j t k j ≤ ∂ t w k i t k i − ∂ t w k j t k j L 2 Ω w k i t k i − w k j t k j L 2 Ω f w k i t k i v − f w k j t k j v L σ Ω w k i t k i − w k j t k j L σ Ω .

4.7
We then conclude form 4.6 that {w k i t k i } is a Cauchy sequence in H 1 0 Ω , and thus {S 1 t } t≥0 is asymptotically compact in L r Ω ∩ H 1 0 Ω , 1 ≤ r < ∞. Using Lemma 2.7, we conclude that {S 1 t } t≥0 possesses a global attractor A v , which is compact, invariant in L r Ω ∩ H 1 0 Ω , and attracts every bounded initial sets of L 1 Ω in the topology of L r Ω ∩ H 1 0 Ω .

Completion of the Proof of Theorem 1.1
Note that

16
Abstract and Applied Analysis Thus we have 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 L 1 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 S 1 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 S 1 t , we know that w t converges to 0 in H 1 0 Ω as t → ∞. Else, we have a sequence t n → ∞, such that C > w t n H 1 0 Ω ≥ > 0. Since the semigroup S 1 t is asymptotically compact, there is a subsequence t n j → ∞, such that w t n j converges to a function χ in H 1 0 Ω and hence in L 1 Ω . Thus χ 0. A contradiction!