Abstract and Applied Analysis

Volume 2012 (2012), Article ID 463082, 15 pages

doi:10.1155/2012/463082

Research Article

Complicated Asymptotic Behavior of Solutions for Heat Equation in Some Weighted Space

Liangwei Wang¹ and Jingxue Yin²

¹School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou 404000, China

²School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

Received 5 June 2012; Accepted 29 July 2012

Academic Editor: Toka Diagana

Copyright © 2012 Liangwei Wang and Jingxue Yin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We investigate the asymptotic behavior of solutions for the heat equation in the weighted space . Exactly, we find that the unbounded function space with can provide a setting where complexity occurs in the asymptotic behavior of solutions for the heat equation.

1. Introduction

In this paper, we consider the asymptotic behavior of solutions to the Cauchy problem of the heat equation

where and the initial value .

Whether complexity occurs in the asymptotic behavior of solutions for some evolution equations or not mainly depends on the work spaces that one selects [1–9]. In the space with , the problem (1.1) under consideration is well posed and the asymptotic behavior of the solutions is rather simple, reflecting the simple structure of the heat equation. Considering, for instance, the problem (1.1) with the initial value , it is well-known that the solutions converge as toward a multiple of the fundamental solution, the one which has the same integral,

where and , see [10, 11].

It was first found in 2002 [12] by Vázquez and Zuazua that the bounded function space provides a setting where complicated asymptotic behavior of solutions may take place for the heat equation. In fact, they proved that, for any bounded sequence in , there exists an initial value and a sequence as such that

uniformly on any compact subset of as . Subsequently, Cazenave et al. showed that, in the bounded continuous function space , the solutions of the heat equation may present more complex asymptotic behavior [13–15]. Meanwhile, considerable attention has also been paid to study the complicated asymptotic behavior of solutions for the porous medium equation and other evolution equations in some bounded function spaces such as and (see, e.g., [3, 7, 9, 12, 16] and the references therein).

In this paper we find that, even in the unbounded function space with , the complicated asymptotic behavior of solutions for the heat equation can also occur. For this purpose, we need to establish the – smoothing effect and other estimates for the solutions of the problem (1.1) when the initial value with or .

The rest of this paper is organized as follows. In the next section, we give some definitions and some estimates of the solutions to the problem (1.1). Section 3 is devoted to study the complicated asymptotic behavior of the solutions.

2. Main Estimates

In this section, we investigate some properties of solutions for the problem (1.1) when the initial value belongs to some weighted spaces. For these purposes, we first introduce the mild solutions of the problem (1.1) which are defined as

Letting and , we define two weighted spaces and as follows:

Endowed with the obvious norms,

the spaces and are both Banach spaces. Notice that if , then

Next we give the definition of the -limit set which is our main study object in this paper.

Definition 2.1. Let , ,  , and suppose that . The -limit set is given by

Here for and .

In the rest of this section, we will consider the properties of the solutions for the problem (1.1) when the initial value or .

The following theorem can be seen as some extension of the maximum principle for the problem (1.1).

Theorem 2.2. Let . Suppose that

and that are the mild solutions of the problem (1.1). Then

Moreover, if , then

or if , then

Remark 2.3. Let . From Theorem 2.2, we can obtain the well-known result (maximum principle) that if , then

Proof. To prove this theorem, we need the fact that if

then there exists a constant such that

which proof can be found in [17]; we give the proof here for completeness. Consider the following problem:

For , from (2.1), we can get that

By the existence and the regularity theories of the solutions, we can obtain that, for ,

see [10, 18]. Now taking , and in the expression (2.14), we have

The fact that clearly implies that, for ,

as . Let

So,

Therefore,

as . So, there exists constant such that

By (2.16), we thus have

Notice that

Therefore, by comparison principle and (2.22), we can get that, for all , there exists constant such that

So we complete the proof of (2.12). For any , from (2.1) and (2.12), we thus obtain that

Therefore, if , then

This clearly illustrates (2.8). If , then

From this, we can get (2.9). So we complete the proof of this theorem.

Theorem 2.4 (– smoothing effect). Let and . Suppose and that are the solutions of the problem (1.1). Then

Moreover, if , then

or if , then

Remark 2.5. If , then Theorem 2.4 captures the result – smoothing effect for the heat equation.

Proof. For any , from (2.1) and Theorem 2.2, we thus obtain that

Here . From this, we can get that, if , then

or if , then

So we complete the proof of this theorem.

In the following theorem, we consider the property of the solutions of (1.1) with the initial data .

Theorem 2.6. Let . If , then

Proof. For , the above theorem is a well-known result. So, in the rest of this proof, we assume that . From (2.1), we have

For any , from , we obtain that there exists an such that if , then

So, from (2.12), we have

So, if , then

or if , then

Notice also that

So, there exists a constant such that

This means that

So, there exists an such that if , then

Combining this with (2.35), (2.38), and (2.39), we can obtain that, for ,

So we complete the proof of this theorem.

3. Complicated Asymptotic Behavior

In this section, we investigate the asymptotic behavior of solutions for the problem (1.1) and give the fact that the weighted space with can provide a setting where complexity occurs in the asymptotic behavior of solutions.

Theorem 3.1. Let , , , and . If

then there exists an initial value and a sequence as such that

That is, .

Proof. Suppose that is a constant. Then let

Now we define the initial-value as

Let

So,

Here we have used the fact that if , then

Therefore, the sequence of (3.4) is convergent in . This means that

As a result of (2.1), we see that, for ,

where

Notice that, if , then

we thus obtain that

Consequently, for any , we can select large enough to satisfy that if , then

Here we have used the hypothesis . So, (2.30), (3.11), and (3.12) indicate

From (3.3), we thus obtain that

as . From (3.7) and the definition of , we have

Applying (2.9) to , we thus have

as . Here we have used (3.3) and the fact that for . At present, we want to verify the claim that, for ,

In fact, if , then

Notice that

Therefore, from Theorem 2.6, we have

So, for any , from (2.35), (2.39), and (2.43), we get that if , there exists a constant independing on and such that

The fact means that

Therefore, for any , there exists an such that if , then

So, from (3.19), (3.22), and (3.24), we thus obtain that

as . From (3.9), (3.15), (3.17), and (3.18), we obtain that for any fixed ,

Equation (3.2) follows from (3.26) by setting and . The proof of this theorem is complete.

Theorem 3.2. Let , , and . If

then there exists an initial value such that

Remark 3.3. If , the above results had been given by Cazenave et al. (see [15, Corollary 6.3]). So our results capture their results. Here we have used some of their ideas.

Proof. By (3.27), there exists a constant such that

Therefore, there exists a countable dense subset of such that

So, there exists a sequence such that

(i)for any , there exists a subsequence of the sequence satisfying

(ii)there exists a constant satisfying

Now we select a constant

and then take

The initial-value is given by

From (3.7), (3.32), and (3.34), we have

This means that the sequence (3.35) is convergent in . So,

Similar to the proof of Theorem 2.2, we can prove that, for any and , there exists a sequence as such that

as . Let and in (2.1); we thus have

Therefore,

Notice that is dense in and that is a closed subset of . We thus obtain from (3.40) that

So we complete the proof of this theorem.

Remark 3.4. For any , there exist constants and such that

Therefore, from Theorem 3.2, we can get that the weighted space provides a setting where complexity occurs in the asymptotic behavior of solutions for the problem (1.1).

Acknowledgment

This work is supported by the National Natural Science Foundation of China, the Specialized Research Fund for the Doctoral Program of Higher Education of China, the Scientific and Technological Projects of Chongqing Municipal Commission of Education.

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