Grow-up rate of solutions for the heat equation with a sublinear source
© Wang and Yin; licensee Springer 2012
Received: 13 June 2012
Accepted: 20 August 2012
Published: 31 August 2012
In this paper, we investigate the grow-up rate of solutions for the heat equation with a sublinear source. We find that if the initial value grows fast enough, then it plays a major role in the growing up of solutions, while if the initial value grows slowly, then the sublinear source prevails. As a direct application of these results, we show that the effect of the sublinear source is negligible in the asymptotic behavior of solutions as if the initial value grows fast enough.
Here , , and with .
see also . For the Cauchy-Dirichlet problem of (1.1), the existence of growing up global solutions and the grow-up rate of solutions has been investigated by Dold, Galaktionov, Lacey and Vázquez in , Galaktionov and King in . If , there are also a lot of papers which intensely investigate the solutions of (1.1)-(1.2) converging to zero at different algebraic rates [11–16].
then the solutions of (1.1)-(1.2) are global.
for large t. Here if , and for any if . Moreover, as an application of these results, we get that if and the initial value satisfies (1.3), (1.4), then the effect of the sublinear source is negligible in the asymptotic behavior of solutions as . While for , Aguirre and Escobedo  revealed that the effect of the sublinear source cannot be negligible in the asymptotic behavior of the solutions as . For the absorption case ( is replaced by in (1.1)) and the supercritical case ( in (1.1)), some similar results about the asymptotic behavior of solutions for these problems were established by a lot of papers, see [18–20].
The paper is organized as follows. The next section is devoted to giving the grow-up rate for the solutions of the problem (1.1)-(1.2) with . In Section 3, we investigate the asymptotic behavior of solutions for the problem (1.1)-(1.2).
2 Growth-up rate of solutions
So, for any , those two growing up effects given by (2.1) and (2.2) can be compared as . When , the one given by (2.2) prevails; when , the one given by (2.1) prevails; and they coincide in the critical case .
If the initial value , the existence and uniqueness of a mild solution for the problem (1.1)-(1.2) has been given in .
Lemma 2.1 ()
for a.e. .
Moreover, if , the convergence is uniform on compact subsets of .
Our results about the grow-up rate of solutions are the following two theorems.
Here is the solution of (1.1)-(1.2).
From this, we get (2.7) easily. So we complete the proof of this theorem. □
In the following theorem, we consider the grow-up rate for the solutions of the problem (1.1)-(1.2) when the nonnegative initial value with .
Here is also the solution of (1.1)-(1.2).
Combining this with (2.14), we can get the desired results. So we complete the proof of this theorem. □
Remark 2.1 From Theorem 2.1 and Theorem 2.2, we find that if , then the main effect on the growing up of solutions comes from the initial value; while if , then the sublinear source has a major effect.
3 Asymptotic behavior
for any which vanishes for large and at .
The following result gives the fact that if , then the sublinear source is negligible in the asymptotic behavior of the rescaled solution as . Similar to [19, 22, 23], we follow the framework by Kamin and Peletier  to give the proof of our result.
uniformly on sets , . Here is the mild solution of (1.1)-(1.2) and .
uniformly on any compact sets of as . By (3.8), we thus have (3.2). So we complete the proof of this theorem. □
By (A.1) and the comparison principle, we have (A.2). So we complete the proof of this lemma. □
The authors are grateful to the referees for their useful comments. This work are supported by NSFC, the Research Fund for the Doctoral Program of Higher Education of China, the Natural Science Foundation Project of “CQ CSTC”, the Scientific and Technological Projects of Chongqing Municipal Commission of Education.
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