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Blow-up and global existence for the non-local reaction diffusion problem with time dependent coefficient
Boundary Value Problems volume 2013, Article number: 239 (2013)
Blow-up and global existence for the non-local reaction diffusion problem with time dependent coefficient under the Dirichlet boundary condition are investigated. We derive the conditions on the data of problem (1.1) sufficient to guarantee that blow-up will occur, and obtain an upper bound for . Also we give the condition for global existence of the solution.
In this work, we consider the following non-local reaction diffusion problem with time dependent coefficient under the Dirichlet boundary condition
where is a bounded domain with a smooth boundary ∂ Ω, Δ is the Laplace operator, and is the possible blow-up time. By the maximum principle, it follows that in the time interval of existence. The coefficient is assumed to be nonnegative. The particular case of of problem (1.1) has already been investigated by many authors, in [1, 2], they studied the question of blow-up for the solution, and in [3–5], they derived lower bounds for blow-up time under different boundary conditions. To deal with problem (1.1) with time dependent coefficient, we make the assumption on the parameters p and q, that is, .
The motivation of this article comes from the work of Payne and Philippin in , where they investigated the blow-up phenomena of the solution of the following problem
where Ω is a bounded sufficiently smooth domain in , , and the coefficient is assumed nonnegative or strictly positive depending on the situation.
In next, we employ a method used by Kaplan in  to obtain a condition, which leads to blow-up at some finite time and also leads to an upper bound for the blow-up time. In Section 3, we derive the condition on the data of problem (1.1) sufficient to guarantee the global existence of .
2 Conditions for blow-up in finite time
Let be the first eigenvalue, and let be the associated eigenfunction of the Dirichlet-Laplace operator defined as
Let the auxiliary function
defined in , where is the solution of (1.1) and .
We assume that for all ,
for some constant β, and
We deduce from (2.4) and (2.5) that
Furthermore, using (2.2) and Hölder’s inequality, we get
Combining (2.6) and (2.7), we get
By integrating (2.8), we get
If for some , then blows up at time . This result is summarized in the following theorem.
Theorem 1 Let be the solution of problem (1.1). Then the auxiliary function defined in (2.3) blows up at time with
3 Condition for global existence
In this section, our argument makes use of the following Sobolev-type inequality
valid in for a nonnegative function v that vanishes on ∂ Ω. In this section, our results restricted to for proof of (3.1), see .
We consider the auxiliary function defined as
we assume that for all ,
for some constant β. In (3.2)-(3.3), n is subjected to restrictions
For convenience, we set
due to (3.4), we obtain
By using Hölder’s inequality,
and Sobolev-type inequality (3.1), we obtain
where Γ is defined in (3.1). Joining (3.11) and (3.9), we obtain
with arbitrary . Choosing , the first eigenvalue of problem (2.1), we have
by the Rayleigh principle. By using (3.13) in the last factor of (3.12), we obtain
Suppose that β is small enough to satisfy the condition
and that initial data is small enough to satisfy the condition
Then either remains negative for all time, or there exists a first time such that
Then we obtain the differential inequality
Integrating this differential inequality, we obtain
This result is summarized in the next theorem.
Theorem 2 Let Ω be a bounded domain in , and assume that the data of problem (1.1) satisfy conditions (3.4), (3.16), (3.17). Then the auxiliary function defined in (3.2) satisfies (3.20), and exists for all time .
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The authors would like to express sincere gratitude to the referees for their valuable suggestions and comments on the original manuscript. This work is supported in part by the NSF of PR China (11371384) and in part by the Natural Science Foundation Project of CQ CSTC (2010BB9218).
The authors declare that they have no competing interests.
All authors contributed to each part of this study equally and read and approved the final version of the manuscript.