Abstract
Keywords:
degenerate parabolic equation; localized source; uniform blowup rate1 Introduction
In this paper, we consider the following parabolic equation with nonlocal and localized reaction:
where Ω is an open ball of , with radius R, and .
Many of localized problems arise in applications and have been widely studied. Equations (1.1)(1.3), as a kind of porous medium equation, can be used to describe some physical phenomena such as chemical reactions due to catalysis and an ignition model for a reaction gas (see [13]).
As for our problem (1.1)(1.3), to our best knowledge, many works have been devoted to the case (see [47]). Let us mention, for instance, when , blowup properties have been investigated by Okada and Fukuda [7]. Moreover, they proved that if and is sufficiently large, every radial symmetric solution (maximal solution) has a global blowup and the solution satisfies
in all compact subsets of Ω as t is near the blowup time , where and are two positive constants. Souplet [4,8] investigated that global blowup solutions have uniform blowup estimates in all compact subsets of the domain.
The work of this paper is motivated by the localized semilinear problem
with Dirichlet boundary condition (1.2) and initial condition (1.3). In the case of and , the uniform blowup profiles were studied in [5,9] and [10], respectively.
It seems that the result of [5,9,10] can be extended to and are two functions. Motivated by this, in this paper, we extend and improve the results of [5,9,10]. Our approach is different from those previously used in blowup rate studies.
In the following section, we establish the blowup rate and profile to (1.1)(1.3).
2 Blowup rate and profile
Throughout this paper, we assume that the functions , and satisfy the following two conditions:
(A1) , and ; , and are positive in Ω.
(A2) , and are radially symmetric; , and are nonincreasing for .
Theorem 2.1Suppose thatsatisfies (A1) and (A2). If, then the solutions of (1.1)(1.3) blow up in finite time for large initial data.
The proof of this theorem bears much resemblance to the result in [7,11,12] and is, therefore, omitted here.
Next we will show that in the situation of localized source dominating (), problem (1.1)(1.3) admits some uniform blowup profile.
Theorem 2.2Assume (A1) and (A2). Letbe the blowup solution of (1.1)(1.3) andis nondecreasing in time. If, then we have
uniformly in all compact subsets of Ω.
Throughout this paper, we denote
In our consideration, a crucial role is played by the Dirichlet eigenvalue problem
Denote by λ the first eigenvalue and by φ the corresponding eigenfunction with in Ω, normalized by . In the following, C is different from line to line. Also, we will sometimes use the notation for with the blowup time for (1.1)(1.3).
In order to prove the results of Section 2, first we derive a fact of the following problem:
Lemma 2.1Assume (A1), (A2) and. Letbe the blowup solution of (2.2) and assume thatis nondecreasing in time, we then have
uniformly in all compact subsets of Ω.
Proof Assumption (A2) implies and on . From (2.2), we have
which implies
Set , and , . By , we obtain that , for .
Introducing a function
In the following, we only consider . For the case of , the proof is similar.
A series calculation yields
In addition, note
Now, according to (2.5) and (2.6), it follows that
Set , , . Since and note that , then there exists such that .
Therefore, in view of (2.7), we observe
Clearly, is a supsolution of the following equation
where in and with . Here we also assume that is a symmetric and nonincreasing function of ().
By the maximum principle, we have and in for .
Similar to the proof of Theorem 3.1 in [9] that
uniformly in all compact subsets of Ω.
By the arbitrariness of , we obtain that the following limit converges uniformly in all compact subsets of
In particular,
This inequality and (2.4) infer that
Multiplying both sides of (2.2) by φ and integrating over , we have, for ,
Since and , it then follows that
Next we prove that
uniformly in any compact subsets of Ω.
Assume on the contrary that there exists () such that
Then there exists a sequence , such that
Using the continuity of , we see that there exists () such that for . Note that and (2.9), we obtain
This contradicts (2.11) and we then complete the proof of Lemma 2.1. □
Lemma 2.2Under the assumption of Theorem 2.2, letbe the blowup solution of (1.1)(1.3), then it holds that
uniformly in all compact subsets in Ω.
Proof Proceeding as in (2.4), we have
Now, according to , it then follows that is a subsolution of the following equation:
By the maximum principle, and in . Using Lemma 2.1, it holds that
uniformly in all compact subsets of Ω.
Hence
uniformly in any compact subsets of Ω, which implies
Combining (2.13) with (2.15), we deduce that
Multiplying both sides of (1.1) by φ and integrating over , we find, for ,
Since and , it then follows that
Therefore,
By analogy with the argument taken in Lemma 2.1, we complete the proof of this lemma. □
Proof of Theorem 2.2 By Lemma 2.2, we infer that
hence
Integrating this equivalence between t and , we obtain
The result finally follows by returning (2.17) to (2.12). □
Remark 2.1 It seems that in the case of , the blowup rate remains valid in all compact subsets, but we do not know how to treat it. (It is an open problem in this case.)
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All the authors typed, read and approved the final manuscript.
Acknowledgements
This work was supported by the China Postdoctoral Science Foundation Founded Project (2013M540405), the National Natural Science Foundation of China (61374194), and the Scientific Innovation Research of College Graduate in Jiangsu Province (CXZZ_0163).
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