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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 312536, 17 pages
Asymptotic Behavior of Approximated Solutions to Parabolic Equations with Irregular Data
1School of Mathematical Sciences, Anhui University, Hefei 230039, China
2School of Traffic and Transportation, Lanzhou Jiaotong University, Lanzhou 730070, China
Received 10 July 2012; Accepted 17 August 2012
Academic Editor: Sergey V. Zelik
Copyright © 2012 Weisheng Niu and Hongtao Li.
Let be a smooth bounded domain in . We consider the asymptotic behavior of solutions to the following problem , where , 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.
We consider the asymptotic behavior of solutions to the following equations where is a bounded domain in with smooth boundary , , , is a finite Radon measure independent of time, is a matrix with bounded, measurable entries, and satisfying the ellipticity assumption Concerning the nonlinear term, we assume that is a function satisfying, both for all , where are positive constants.
Parabolic equations with 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 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 , where the authors obtained weak solutions (in distribution sense) to nonlinear parabolic equations. In our setting, such a solution is a function such that for any , and for any .
Generally, the regularity of weak solutions (in distribution sense) is not strong enough to ensure uniqueness . 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 . 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 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 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 , to ensure the existence result for large in (1.5) , we restrict ourselves to diffuse measures, that is, does not charge the sets of zero parabolic 2-capacity (see details for parabolic -capacity in ). 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 in with , which attracts every bounded subset of in the norm of , for any .
For the case , we consider general bounded Radon measure 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 , for any , though they all lie in some less regular spaces.
Our main results can be stated as follows.
Theorem 1.1. Assume that , , is a bounded Radon measure, which does not charge the sets of zero parabolic 2-capacity and is independent of time, is a function satisfying assumptions (1.3)–(1.5). Then the semigroup , generated by approximated solutions of problem (1.1), possesses a global attractor in . Moreover, is compact and invariant in with , and attracts every bounded subset of in the norm topology of , .
Theorem 1.2. Assume that , , is a bounded Radon measure independent of time. Then the approximated solution of problem (1.1) is unique and converges to the unique approximated solution of the corresponding elliptic equations in the norm topology of , for any .
Remark 1.3. Though and all lie in some less-regular spaces, converges to in stronger norm, that is, converges to in , . Such a result, in some sense, sharpens the result of , where the author showed that converges to in .
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 we use to denote hereafter. Also, we denote by the Lebesgue measure of the set , and denote by 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 is called an approximated solution of problem (1.1), if , for any , and for any , and moreover, is obtained as limit of solutions to the following approximated problem where is a smooth approximation of data .
Proof. According to [18, Theorem 2.12], if a Radon measure on does not charge the sets of zero parabolic 2-capacity and is independent of time, can actually be identified as a Radon measure which is absolutely continuous with respect to the elliptic 2-capacity. Using Theorem 2.1 of , can be decomposed as , where . Hence, we need only to consider the following problem
The proof of existence part of the theorem is similar to . Besides, one can prove using arguments similar to CLAIM 2 in . So we omit the details of them and only prove the uniqueness result.
Let , be a smooth approximation of data and with and let , be another smooth approximation of the data with Assume that are two approximated solutions to problem (1.1), obtained as limit of the solutions to the following two approximated problems, respectively, Now we prove that . For any , define as Let be its primitive function. Taking as a test function in (2.6) and (2.7), we deduce that Hence, from the assumptions on , we get Let , we have Thus for all , we have Taking small enough such that , we deduce that for all in , thus in . Dividing into several intervals to carry out the same arguments, we obtain the uniqueness of the approximated solution.
Similar to , we can prove the following.
Theorem 2.3. Under the assumptions of Theorem 1.1, there exists at least one approximated solution to the stationary problem of corresponding to problem (1.1), with (i)if , we have for ;(ii)if , we have for .
Remark 2.4. Note that if is an approximated solution to following problem then there is a sequence converges to , where is the solution of the corresponding approximated problem And hence is a solution of parabolic equations Thus, is an approximated solution of problem (1.1) with initial data .
Under the assumptions of Theorem 1.2, the problem turns out to be The existence of approximated solutions to problem (1.1) and the elliptic equations corresponding to it follows directly from Sections and of . Form Section of , 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.
Proof. The proof is mainly similar to that of Theorem 6 in . We just sketch it. Let be an approximated solution, then there exist a smooth approximation , of data and , such that the solution of the approximated problem of (2.16) with data and converges to . Let and be the solution of the the following parabolic problem where is the transposed matrix of . Taking as a test function in the approximated problem and taking as a test function in the problem above, then let go to infinity we obtain that the approximated solution is a duality solution. Form the uniqueness of duality solutions , 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 ). Let , be two Banach spaces, let be separable, reflexive, and let with dual . Suppose that is uniformly bounded in with and that weakly in for some . Then Moreover, if , then in fact
Lemma 2.7 (see ). Let be two Banach spaces with imbedding , let be a continuous semigroup on . Assume that is asymptotically compact in and has an absorbing set , that is, for any bounded set , there exists a such that Then has a global attractor in , which is compact, invariant in and attracts every bounded sets of in the topology of .
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 , we need neither of them. We only need the semigroup to be continuous in a less-regular space .
3. Improved Regularity Results on the Approximated Solutions
In this section, we prove the following regularity results on the approximated solution to problem (1.1).
Theorem 3.1. Under the assumptions of Theorem 2.2, let be the approximated solution to problem (1.1). Then admits the decomposition , with being an approximated solution to problem (2.13), and being an approximated solution of the following problem Moreover, we have (i) for any . Moreover, there exists a constant and a time such that .(ii) for any . Moreover, there exists a constant and a time such that .
Proof. We follow the lines of [15, 28]. Let be a sequence of smooth data which converges to in and . Let be a solution of the following approximated problem for each ,
Then converges (up to subsequences) to an approximated solution strongly in , and weakly in . Let be a sequence of solutions to the following approximated problem
where converges to with . Similar to [8, 29], we know that
Now let . Then satisfies
Similarly, we have (up to subsequences) converges to the approximated solution of problem (3.1) in and weakly in .
Now we prove (i). Taking as test function in (3.3) (for simplicity we take ), we deduce that Since we have The Gronwall's inequality implies that Noticing that we obtain that Moreover, integrating (3.6) between and and using (3.7) we have Similarly, taking as test function in (3.2), we can deduce that Hence, with independent of , for .
Now we use bootstrap method in the case . The case can be treated similarly with minor modifications. Multiplying (3.5) by , and integrating on , we obtain Since , we deduce that Integrating (3.16) between and , it yields Integrating the above inequality with respect to between and , we get Therefore, Integrating (3.16) on for , we deduce that Note that (3.20) insures that, for any , there exists at least a such that Standard Sobolev imbedding implies that Now multiplying (3.5) by , we have Using Hölder inequality, and Young inequality we deduce that Taking (3.24) into (3.23), it yields Integrating (3.25) between and , we have Therefore, from (3.19) and (3.22) we get with independent of . Integrating (3.25) between and for , we obtain Similar to (3.22), for any , there exists at least a such that Bootstrap the above processes, we can deduce that with , and independent of . Note that in and . From Lemma 2.6, we have Taking large enough, we get the second part of (i) proved. If the integration are taken over instead of , we get the first part of (i).
Now we are in the position to prove (ii). We multiply (3.5) with and deduce that integrating over , we get with independent of . Now, multiplying (3.5) with , we obtain where . Integrating (3.34) between and gives Now, integrating the above inequality with respect to between and we have Since We deduce that From the assumption (1.4) on , we have Using results in (3.13) and (3.30), we know that Set . Combining (3.33), (3.34), (3.36), and (3.40) we have From Lemma 2.6, we obtain Thus we get the second part of (ii) proved. Taking integration over instead of , the first part of (ii) follows. The proof is completed now.
4. Proof of the Main Theorems
Let be the semigroup generated by problem (1.1) and let be an approximated solution to problem (2.3). Define Then it is easy to verify that is a continuous semigroup in and hence in . From the results in Section 3, we know that the semigroup possesses a global attractor in . 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 possesses a global attractor in , that is, is compact, invariant in , and attracts every bounded initial set of in the norm topology of .
Proof. From Theorem 3.1, we know that possesses an absorbing set in . Also possesses absorbing sets , , respectively, in and for any .
In the next, we prove the asymptotic compactness of . Before that we establish the following estimate Actually, differentiating (3.5) in time and denoting , we have Multiplying (4.3) by and using (1.3), we deduce that Integrating the above inequality between and gives Integrating the above inequality with respect to between and , using (3.42) we obtain Now we prove the asymptotic compactness of the semigroup , that is, for any sequences , sequence has convergent subsequences, where . Since is compact in , there is a subsequence , which is a Cauchy sequence in . Denoting , we deduce that We then conclude form (4.6) that is a Cauchy sequence in , and thus is asymptotically compact in .
Using Lemma 2.7, we conclude that possesses a global attractor , which is compact, invariant in , and attracts every bounded initial sets of in the topology of .
Proof of Theorem 1.2. Let 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 converges to in as . Using arguments similar to Section 3, we can prove similar regularity results for and then prove the asymptotic compactness of the semigroup as in Theorem 4.1. Thus, we obtain that converges to in , as . Moreover from the asymptotic compactness of the semigroup , we know that converges to in as . Else, we have a sequence , such that . Since the semigroup is asymptotically compact, there is a subsequence , such that converges to a function in and hence in . Thus . A contradiction!
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).
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