Abstract and Applied Analysis

Volume 2014 (2014), Article ID 329474, 4 pages

http://dx.doi.org/10.1155/2014/329474

Research Article

Algebraic Decay for Weak Solutions of the Nonlinear Heat Equations in Whole Space R ³

Yuexing Yang

Department of Mathematics, School of Science, Wuhan University of Technology, Wuhan 430070, China

Received 13 May 2014; Accepted 18 June 2014; Published 1 July 2014

Academic Editor: Bo-Qing Dong

Copyright © 2014 Yuexing Yang. 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 obtained the algebraic time decay rate for weak solutions of the nonlinear heat equations with the nonlinear term in whole space . The methods are based on energy methods and Fourier analysis technique.

1. Introduction

In this study we consider the Cauchy problem of the following nonlinear heat equations in whole spaces:

with the initial data

Here, is the unknown function and

This nonlinear model appears to be relevant in the theory of the viscous incompressible Newtonian and non-Newtonian fluid flows (refer to [1–4] and the references therein). As for the generalized nonlinear heat equations

there is much study on the well-posedness and large time behavior for solutions to these nonlinear heat equations [5–7]. For example, Ponce [8] and Zheng and Chen [9] studied global weak solutions of the generalized nonlinear heat equations by the difference methods. Escobedo and Zuazua [10] and Zhang [11, 12] studied the time decay for weak solutions of the nonlinear heat equation with filtration-type nonlinear terms. Zhao [13] recently examined asymptotic behavior for weak solutions of the nonlinear heat equations with the nonlinear term satisfying some growth conditions. One may also refer to some interesting time decay results for weak solutions of the nonlinear partial differential equations models [1, 14, 15]. More precisely, the optimal decay estimates for solutions of those nonlinear models such as

Motivated by the results [16, 17], the main purpose of this study is to investigate the algebraic time decay for weak solutions of the nonlinear heat equation (1)-(2). More precisely, when the initial data satisfies some integrable condition, we will show the weak solution of the nonlinear heat equation (1)-(2) decays as that of the linear heat equation. The result is proved by developing some analysis techniques, such as Fourier transformation and energy inequality.

2. Statement of the Main Results

In this section we first introduce some notations. We denote by with the usual Lebesgue space with the norm . In particular, . is the usual Sobolev space with the norm (refer to [18]). is any constant which may only depend on the initial data but never depend on . The Fourier transformation of is or ,

For a weak solution of the nonlinear heat equation (1)-(2), we mean a function

which satisfies

and energy inequality

Our main result reads as follows.

Theorem 1. Let ; then the weak solution of the nonlinear heat equation (1)-(2) decays as

Remark 2. It is well known that the solution of the linear heat equation

has the same time decay properties as

with the initial data . Therefore our result cannot be improved under the same condition on initial data. Moreover, it seems as an interesting and important problem to further consider the explicit error estimates between the above nonlinear heat equation and the linear heat equation; we will study this issue in the forthcoming paper.

Remark 3. Compared with the previous results on the time decay issue of the nonlinear heat equations, the main difficulty here lies in the uniform estimates of the nonlinear term . In order to overcome this difficulty, we further develop the classic so-called Fourier splitting methods which are first introduced by Schonbek [16] when she studied the time decay of Navier-Stokes equations. And then we further need to apply Hölder inequality, Young inequality, and energy inequality, together with the rigorous computation.

3. Proof of Theorem 1

In order to prove Theorem 1, the existence of weak solution of the nonlinear heat equation can be derived by the standard methods (refer also to [19], e.g.); we first deduce some auxiliary lemmas which will be employed in the proof of the main theorems.

Lemma 4. Under the same condition in Theorem 1, one has

Proof of Lemma 4. We first take Fourier transformation to both sides of the nonlinear heat equation (1)

We can look at the above equation (14) as a linear ordinary differential equation with respect to , and according to the fundamental theory of the ordinary differential equation, it is easy to check that the solution of (14) can be expressed as

Now we need to estimate the right-hand side of (15). Since

we have used Hölder inequality and Young inequality.

Moreover, integrating the energy inequality (9) in time from to gives

which, together with , yields

Plugging (18) into (15) gives that

here we have used the fact .

Hence, we complete the proof of Lemma 4.

Lemma 5 (Parseval equality [7]). If , then the Fourier transformation of satisfies and

Now we begin to prove Theorem 1.

From the energy inequality (9), we have

With the aid of Parseval equality, we rewrite (21) as

Choosing a smooth function , such that

and then multiplying with both sides of the inequality (22), one shows that

Let

then we have

Inserting the above inequality into (24), it follows that

and integrating in time yields

For , by means of the initial condition in Theorem 1 and Parseval equality, it is easy to check that

For , by employing Lemma 4 and direct computation, we have

We now choose to get

Thus,

Inserting the estimates of and into (28) becomes, note that

from which, and together with, Parseval equality implies that

Hence we complete the proof of Theorem 1.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is partially supported by the Innovation Fund of Wuhan University of Technology, no. 2013-la-024.

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