Research Article | Open Access

Liangwei Wang, Jingxue Yin, "Complicated Asymptotic Behavior of Solutions for Heat Equation in Some Weighted Space", *Abstract and Applied Analysis*, vol. 2012, Article ID 463082, 15 pages, 2012. https://doi.org/10.1155/2012/463082

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

**Academic Editor:**Toka Diagana

#### 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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#### Copyright

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.