Extinction and decay estimates of solutions for a porous medium equation with nonlocal source and strong absorption
© Xu et al.; licensee Springer 2013
Received: 25 September 2012
Accepted: 21 December 2012
Published: 5 March 2013
In this paper, we investigate extinction properties of the solutions for the initial Dirichlet boundary value problem of a porous medium equation with nonlocal source and strong absorption terms. We obtain some sufficient conditions for the extinction of nonnegative nontrivial weak solutions and the corresponding decay estimates which depend on the initial data, coefficients, and domains.
where , , () is a bounded domain with smooth boundary, and is a nonnegative function. The symbols and , where , denote - and -norm, respectively, and denotes the measure of Ω.
Equation (1) describes the fast diffusion of concentration of some Newtonian fluids through a porous medium or the density of some biological species in many physical phenomena and biological species theories, while nonlocal source and absorption terms cooperate and interact with each other during the diffusion. It has been known that the nonlocal source term presents a more realistic model for population dynamics; see [1–3]. In the nonlinear diffusion theory, obvious differences exist among the situations of slow (), fast (), and linear () diffusions. For example, there is a finite speed propagation in the slow and linear diffusion situations, whereas an infinite speed propagation exists in the fast diffusion situation.
subject to (2) and (3). They obtained some conditions for the extinction and non-extinction of solutions to equation (4) and decay estimates by the upper and lower solutions method. On extinctions of solutions to the p-Laplacian equations or the doubly degenerate equations, we refer readers to [13, 14] and the references therein.
and obtained sufficient conditions for the extinction and non-extinction of solutions to that equation. Thereafter, Fang and Li  extended their results to the doubly degenerate equation in the whole dimensional space.
For equation (1) with , , and , Han and Gao  showed that is the critical exponent for the occurrence of extinction or non-extinction. When , , and , the conditions for the extinction and non-extinction of solutions and corresponding decay estimates were obtained (cf. ). Recently, Fang and Xu  considered equation (1) with , when the diffusion term was replaced with a p-Laplacian operator in the whole dimensional space, and showed that the extinction of the weak solution is determined by the competition of two nonlinear terms. They also obtained the exponential decay estimates which depend on the initial data, coefficients, and domains. The extinctions of solutions to equation (1) with nonlocal source terms do not depend on the first eigenvalue of the corresponding operator, which is different from the case of local source terms. The extinction and decay estimates for solutions to the nonlocal fast diffusion equations with nonzero coefficients and strong absorption terms, like equation (1), are still being investigated.
Motivated by the above works, we investigate whether the existence of strong absorption can change extinction behaviors for solutions to problem (1)-(3) in the whole dimensional space. The main tools we use are the integral estimate method and the Gagliardo-Nirenberg inequality. This technique has a wide application, especially for equations that do not satisfy the maximum principle (cf. ). Our goals are to show that the extinction of nonnegative nontrivial weak solutions to problem (1)-(3) occurs when and to find the decay estimates depending on the initial data, coefficients, and domains.
Our paper is organized as follows. In Section 2, we give preliminary knowledge including lemmas that are required in the proofs of our results and present the proofs for the results in Section 3.
2 Preliminary knowledge
- c.For every and every ,
A function u is called a locally weak solution of problem (1)-(3) if it is both a subsolution and a supersolution for some .
Remark 1 The existence and uniqueness of locally nonnegative solutions in time to problem (1)-(3) can be obtained by the standard parabolic regular theory that can be applied to get suitable estimates in the standard limiting process (cf. [2, 21, 22]). The proof is similar to the ones in the cited references, and so it is omitted here.
and using the comparison principle, one can easily obtain the result. □
Lemma 2 (The Gagliardo-Nirenberg inequality) 
where C is a constant depending on N, m, r, j, k, and q such that and . While if , then , and if , then .
3 Main results
In this section, we give some extinction properties of nonnegative nontrivial weak solutions of problem (1)-(3) stated in the following theorems. The corresponding decay estimates to the solutions will be presented in the proofs of the theorems for brief expressions instead of in the statements.
Theorem 1 Suppose that and . Then the nonnegative nontrivial weak solution of problem (1)-(3) vanishes in finite time for any nonnegative initial data provided that either or λ is sufficiently small.
where . Since , and , it can be easily seen that .
Setting , we have .
where , which give the decay estimates in finite time for .
where , which give the decay estimates in finite time for such that .
where . □
Theorem 2 If , then the nonnegative nontrivial weak solution of problem (1)-(3) vanishes in finite time provided that , or λ is sufficiently small, and , where if , then , and if , then .
Since , we have , and hence, if , then .
where . □
Remark 2 Since the Sobolev embedding inequality cannot be used in the proof of Theorem 2, it is not necessary to consider the cases that and , when . In addition, if , the conditions in Theorem 2 imply that .
Theorem 3 Suppose that and . Then the nonnegative nontrivial weak solution of problem (1)-(3) vanishes in finite time for any nonnegative initial data provided that β is sufficiently large.
where . □
Remark 3 One can see from Theorems 1-3 that the extinction of nonnegative nontrivial weak solutions to problem (1)-(3) occurs when .
Remark 4 Theorems 1-3 all require , λ, or to be sufficiently small or β to be sufficiently large.
The second and third authors were supported by the National Science Foundation of Shandong Province of China (ZR2012AM018) and Changwon National University in 2012, respectively. The authors would like to express their sincere gratitude to the anonymous reviewers for their insightful and constructive comments.
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