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Blow-up profile for a degenerate parabolic equation with a weighted localized term
Boundary Value Problems volume 2013, Article number: 269 (2013)
Abstract
In this paper, we investigate the Dirichlet problem for a degenerate parabolic equation . We prove that under certain conditions the solutions have global blow-up, and the rate of blow-up is uniform in all compact subsets of the domain. Moreover, the blow-up profile is precisely determined.
1 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 [1–3]).
As for our problem (1.1)-(1.3), to our best knowledge, many works have been devoted to the case (see [4–7]). Let us mention, for instance, when , blow-up 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 blow-up and the solution satisfies
in all compact subsets of Ω as t is near the blow-up time , where and are two positive constants. Souplet [4, 8] investigated that global blow-up solutions have uniform blow-up estimates in all compact subsets of the domain.
The work of this paper is motivated by the localized semi-linear problem
with Dirichlet boundary condition (1.2) and initial condition (1.3). In the case of and , the uniform blow-up 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 blow-up rate studies.
In the following section, we establish the blow-up rate and profile to (1.1)-(1.3).
2 Blow-up 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 non-increasing for .
Theorem 2.1 Suppose that satisfies (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 blow-up profile.
Theorem 2.2 Assume (A1) and (A2). Let be the blow-up solution of (1.1)-(1.3) and is non-decreasing 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 blow-up 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 2.1 Assume (A1), (A2) and . Let be the blow-up solution of (2.2) and assume that is non-decreasing 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
Thus and .
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
for , .
Set , , . Since and note that , then there exists such that .
Therefore, in view of (2.7), we observe
Set , . We then obtain
Clearly, is a sup-solution of the following equation
where in and with . Here we also assume that is a symmetric and non-increasing 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 2.2 Under the assumption of Theorem 2.2, let be the blow-up 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
which implies and .
Now, according to , it then follows that is a sub-solution 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 blow-up rate remains valid in all compact subsets, but we do not know how to treat it. (It is an open problem in this case.)
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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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Zeng, W., Lu, X., Fei, S. et al. Blow-up profile for a degenerate parabolic equation with a weighted localized term. Bound Value Probl 2013, 269 (2013). https://doi.org/10.1186/1687-2770-2013-269
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DOI: https://doi.org/10.1186/1687-2770-2013-269