1. Introduction

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 [7], 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 [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 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) [17], we restrict ourselves to diffuse measures, that is, does not charge the sets of zero parabolic 2-capacity (see details for parabolic -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 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 [13], 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 .

Theorem 2.2. Under the assumptions of Theorem 1.1, problem (1.1) has a unique approximated solution , for any , satisfying (i) if , then with ;(ii) if , then with .

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 [19], 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 [9]. Besides, one can prove using arguments similar to CLAIM 2 in [8]. 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 [20], 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 [7]. Form Section 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 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 [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 , 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 [16]). 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.