- Open Access
Critical extinction exponents for a nonlocal reaction-diffusion equation with nonlocal source and interior absorption
© Gao and Zheng; licensee Springer. 2014
Received: 10 October 2013
Accepted: 23 December 2013
Published: 16 January 2014
This paper is concerned with a nonlocal reaction-diffusion equation with nonlocal source and interior absorption , , , , , , , . We investigate the critical extinction exponents for the problem based on some adequate supersolutions and subsolutions.
MSC:35K57, 35B33, 35K10.
where is a nonnegative, smooth, symmetric radially function with and supported in the unitary ball, . We assume that is a nonnegative function.
with and and showed that is the critical extinction exponent by invoking the regularizing effect. In this paper under the appropriate hypotheses , we discuss problem (1.1) and obtain the extinction condition by using the principal eigenvalue of the nonlocal heat equation, and thus avoid using the regularizing effect, since there is no regularizing effect in general . It is noted that our approach can be adopted to deal with the blow-up behavior of solutions of nonlocal reaction-diffusion equations with nonlocal source or local source, which was considered in [7, 19].
Now our main results can be stated as follows.
2 Main results
If , then the solution of problem (1.1) vanishes in infinite time for any appropriately small initial data.
If , then the solution of problem (1.1) vanishes in finite time for any conveniently small initial data.
That vanishes in infinite time means that for any .
That vanishes in finite time means that there exists , such that for any and .
If , then problem (1.1) admits at least one nonextinction solution for any nonnegative initial data provided that is appropriately large.
If , then problem (1.1) admits at least one nonextinction solution for any nonnegative initial data provided that λ is appropriately large.
If , then problem (1.1) admits at least one nonextinction solution for any nonnegative initial data.
If , then problem (1.1) admits at least one nonextinction solution for any nonnegative initial data provided that is sufficiently large.
Before proving our main results, we will give some preliminary lemmas, which play a crucial role in the following proofs. As for the proofs of these lemmas, we will not repeat them again.
Applying almost exactly the same arguments as in the proof of Lemma 5 in , we conclude to the following lemma.
where and . Then the above ODE problem has at least one non-constant solution.
Next, our aim is to prove the local existence of solutions to equation (1.1) and the validity of the comparison principle. First, we give the definition of supersolution and subsolution.
where . The subsolution is defined similarly by reversing the inequalities. Furthermore, if u is a supersolution as well as a subsolution, then we call it a solution of problem (1.1).
The existence of the solution of problem (1.1) will be obtained via the successive approximation which comes from .
Lemma 2.2 Let . Then there exists , such that problem (1.1) has nonnegative solutions.
with for all most . Then it follows from the Lebesgue dominated convergence theorem that is the solution of problem (1.1). □
In the following, we conclude that a comparison principle holds for solutions to problem (1.1).
Lemma 2.3 Let , be the supersolution and the subsolution to equation (1.1), respectively. If either and is upper bounded or and has a positive lower bound, then in .
which implies that in . The assertion can be proved similarly for the case and has a positive lower bound. Thus the proof of this lemma is completed. □
Once the existence of the solution to problem (1.1) and the comparison principle are ensured, we begin to analyze the extinction exponents for nonnegative solutions. As a first step we discuss the infinite time extinction of the solution.
Proof of Theorem 2.1 The proof can be divided into two steps:
and then by Step I, we end up with that the solution of equation (1.1) vanishes in infinite time. The proof of this theorem is completed. □
Proof of Theorem 2.2 The proof is similar to that of Theorem 2.1, so we sketch it briefly here. We will prove the theorem in two cases.
Thus, thanks to Lemma 2.3, we derive (), for any fixed . Therefore, , which, together with the arbitrariness of and implies that . Furthermore, setting , then satisfies equation (1.1). According to the above proof, we claim that with any . Now, by virtue of the relation of the extinction time of to , we finally conclude that for any , namely for all .
According to the above results, the solution of equation (1.1) vanishes in finite time. This completes the proof of Theorem 2.2. □
Proof of Theorem 2.3 The proof can be divided into four cases.
which implies w is a subsolution of problem (1.1). Therefore problem (1.1) admits a solution satisfying , which, combined with () implies that is a nonextinction solution of equation (1.1) for any nonnegative initial data provided that is appropriately large.
Since and , we conclude that is a nondecreasing and . Let . Then we can easily derive . Therefore, is a nonextinction solution of equation (1.1) for any nonnegative initial data provided that is appropriately large.
Case IV: If and , employing exactly the same arguments as in the proof of Case I, we finally conclude the result. □
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