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Grow-up rate of solutions for the heat equation with a sublinear source
Boundary Value Problems volume 2012, Article number: 96 (2012)
Abstract
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.
MSC:35K55, 35B40.
1 Introduction
We consider the Cauchy problem of the heat equation with the source
Here , , and with .
After the famous work [1], this problem has been widely studied by several authors. It is well known that any positive solutions blow up in finite time if [1–3], while positive global solutions exist if [1, 4]. Let
If , the existence of growing up global solutions, the solutions exist for any and as in some senses, has been established by Poláčik and Yanagida [5, 6]. If and the initial data satisfy some conditions, Fila, Winkler and Yanagida [7] in 2004 precisely evaluated the grow-up rate of solutions of (1.1)-(1.2) and they found that for large t and some , the solution satisfies
see also [8]. 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 [9], Galaktionov and King in [10]. 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].
For the sublinear case ( in (1.1)), it was Aguirre and Escobedo [17] who first proved that if , and the initial value satisfies
then the solutions of (1.1)-(1.2) are global.
Our interest in this paper is to investigate the grow-up rate of solutions for the problem (1.1)-(1.2) with a sublinear source. We first show that if the initial value satisfies
and
then the solutions of (1.1)-(1.2) () are growing up solutions such that
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 [17] 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
We take in the rest of this paper. For any , we define a weighted space as
with the norm , where . If , then there exist two subsolutions of the problem (1.1)-(1.2):
and
Using a similar method as in [21] (see the Appendix), we can get that there exist constants such that
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 .
Inspired by the above discussions, in this paper we first study the grow-up rate of solutions for the problem (1.1)-(1.2). The mild solution of the problem (1.1)-(1.2) is defined as follows:
If the initial value , the existence and uniqueness of a mild solution for the problem (1.1)-(1.2) has been given in [17].
Lemma 2.1 ([17])
Suppose and , then there exists a unique mild global solution u for the problem (1.1)-(1.2) with such that
-
I.
;
-
II.
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.
Theorem 2.1 Let , and . Suppose
and
Then there exist constants , such that
Here is the solution of (1.1)-(1.2).
Proof The hypothesis (2.6) clearly implies that there exists a constant such that if , then
So,
From the property of the heat semigroup, we have
where . Using a similar method as [21] (see (A.5)), we obtain that there exists a constant such that
So, for , there exists a constant (depending on A and σ) such that
It follows from the comparison principle that
From and I of Lemma 2.1, we obtain that there exists a constant (depending on τ) such that
Therefore,
So, from (2.3), we have
where and are positive constants depending on A, σ and τ. The hypothesis indicates that
Let
So,
Therefore, is an increasing function satisfying
From (2.8), we have
Let , and assume that
So, from (2.10), one can verify that is a supersolution of the following problem:
By (2.8), (2.9) and the comparison principle, we get that
This means that
Let . So, there exist two constants, which we still write as and , such that
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 .
Theorem 2.2 Let , and assume that . If the initial value satisfies (2.5) and (2.6), then for any , there exist constants such that
Here is also the solution of (1.1)-(1.2).
Proof Using the same method as the proof of (2.8), we can get if , then there exist such that
So, by the comparison principle, we can get that there exists a constant satisfying
We first consider the case of . Let
So, is a subsolution of the following problem:
Here we have used the facts that and . Therefore, by the comparison principle, for , there exists a constant C satisfying
From this, we can get that there exist such that
Now, we consider the case of . Let
Then, we define the function
Therefore, is also a subsolution of the problem (2.13). In fact,
and
so,
Using the same method as above, we can get that (2.14) holds for . Without loss of generality, we can assume that in the rest of this proof. From the definition of the mild solutions with (2.12), we have
Here we have used and Lemma A.1, see the Appendix. The assumption implies that
Therefore,
By (2.14), we deduce that there exists a constant C such that
This implies that
Using the integral expression (2.4) again, we have
Here we have used the fact that
Notice that for and ,
where . So,
Iterating (2.15) times, we get that
Here we have used the facts that
and
So, for any , we can select n large enough to satisfy
From (2.16), we thus have
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
In this section, we will use the fact that the mild solutions of the problem (1.1)-(1.2) given by Lemma 2.1 also satisfy the following integral identity:
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 [19] to give the proof of our result.
Theorem 3.1 Let and . If the initial value satisfies (2.5) and (2.6), then
uniformly on sets , . Here is the mild solution of (1.1)-(1.2) and .
Proof We first define the functions
and
Using the comparison principle, we know that
and
For , without loss of generality, we can assume that λ is large enough to satisfy , where T is the constant given by Theorem 2.1. So, from (2.11), we have
So, if and , then
Similarly, for any , from (3.3), we can obtain the following integral estimates:
Using the same method as above and the comparison principle, we can get the similar integral estimates for . For any , from (3.1), we have
where , , and which vanishes for large and at . For any , by the integral estimates of and , there exists such that
From the fact that and (3.4), we can get that there exists such that if , then
Let . From (3.1), we can get that for any compact subset K of , and have a uniform upper bound, which means that the sequence is equicontinuous on K (see [17, 24, 25]). So, we can get that there exist a subsequence and a function such that
as uniformly on K. Therefore, we have as , omitting the primes,
Combining this with (3.5)-(3.7), we obtain that
Therefore, it follows from the uniqueness of the solutions of the heat equation that
Thus the entire sequence converges to . Therefore, we have proved that for any ,
uniformly on any compact subset of . Thus, taking and , we obtain
From (2.8) and , we have
So, for any , by Lebesgue’s dominated convergence theorem, we have
as . The uniform upper bound of on any compact subset M of implies that the sequence is equicontinuous on M, see [17, 24, 25]. Therefore, from (3.9), we have
uniformly on any compact sets of as . By (3.8), we thus have (3.2). So we complete the proof of this theorem. □
Appendix
Lemma A.1 Let and . If
then there exist two constants such that
Proof Consider the following problem:
where is a constant. For , from (2.1), we can get
By existence and regularity theories for solutions, we can obtain that for ,
see [24, 25]. Now taking , and in the expression (A.3), we have
The fact that clearly implies that for ,
as . Let
So
Therefore,
as . So, there exist constants satisfying
By (A.4), we thus have
Let . So there exist two constants such that
Therefore, by the comparison principle and (A.5), we can get that for all , there exist constants such that
By (A.1) and the comparison principle, we have (A.2). So we complete the proof of this lemma. □
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Acknowledgements
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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Wang, L., Yin, J. Grow-up rate of solutions for the heat equation with a sublinear source. Bound Value Probl 2012, 96 (2012). https://doi.org/10.1186/1687-2770-2012-96
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DOI: https://doi.org/10.1186/1687-2770-2012-96