Abstract
In this paper, we investigate the Dirichlet problem for a degenerate parabolic equation
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
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
in all compact subsets of Ω as t is near the blowup time
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
It seems that the result of [5,9,10] can be extended to
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
(A1)
(A2)
Theorem 2.1Suppose that
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 (
Theorem 2.2Assume (A1) and (A2). Let
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 order to prove the results of Section 2, first we derive a fact of the following problem:
Lemma 2.1Assume (A1), (A2) and
uniformly in all compact subsets of Ω.
Proof Assumption (A2) implies
which implies
Thus
Set
Introducing a function
In the following, we only consider
A series calculation yields
In addition, note
Now, according to (2.5) and (2.6), it follows that
for
Set
Therefore, in view of (2.7), we observe
Set
Clearly,
where
By the maximum principle, we have
Similar to the proof of Theorem 3.1 in [9] that
uniformly in all compact subsets of Ω.
By the arbitrariness of
In particular,
This inequality and (2.4) infer that
Multiplying both sides of (2.2) by φ and integrating over
Since
Next we prove that
uniformly in any compact subsets of Ω.
Assume on the contrary that there exists
Then there exists a sequence
Using the continuity of
This contradicts (2.11) and we then complete the proof of Lemma 2.1. □
Lemma 2.2Under the assumption of Theorem 2.2, let
uniformly in all compact subsets in Ω.
Proof Proceeding as in (2.4), we have
which implies
Now, according to
By the maximum principle,
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
Since
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
The result finally follows by returning (2.17) to (2.12). □
Remark 2.1 It seems that in the case of
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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