Abstract
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.
1 Introduction
We consider the initial Dirichlet boundary value problem for a class of porous medium equations with nonlocal source and strong absorption terms
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 [13]. 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.
Recently, many scholars have been devoted to the study of blowup and extinction properties of solutions for nonlinear parabolic equations with nonlocal terms. The blowup rates and blowup sets of solutions to equation (1) have been investigated when , , and the linear absorption term is replaced with a nonlinear term with exponent (cf.[49]). Extinction is the phenomenon whereby there exists a finite time such that the solution is nontrivial for and then for all . In this case, T is called the extinction time. It is also an important property of solutions for nonlinear parabolic equations which have been studied by many researchers. For instance, Evans and Knerr [10] investigated the extinction behaviors of solutions for the Cauchy problem of the semilinear parabolic equation
by constructing a suitable comparison function. Ferreira and Vazquez [11] studied the extinction phenomena of solutions for the Cauchy problem of the porous medium equation with an absorption term
by using the analysis of selfsimilar solutions. Li and Wu [12] considered the problem of the porous medium equation with a source term
subject to (2) and (3). They obtained some conditions for the extinction and nonextinction of solutions to equation (4) and decay estimates by the upper and lower solutions method. On extinctions of solutions to the pLaplacian equations or the doubly degenerate equations, we refer readers to [13,14] and the references therein.
Replacing the nonlocal term in equation (1) with a local term, Liu [15]et al. considered the initial Dirichlet boundary value problem for a class of porous medium equations
and obtained sufficient conditions for the extinction and nonextinction of solutions to that equation. Thereafter, Fang and Li [16] extended their results to the doubly degenerate equation in the whole dimensional space.
For equation (1) with , , and , Han and Gao [17] showed that is the critical exponent for the occurrence of extinction or nonextinction. When , , and , the conditions for the extinction and nonextinction of solutions and corresponding decay estimates were obtained (cf.[18]). Recently, Fang and Xu [19] considered equation (1) with , when the diffusion term was replaced with a pLaplacian 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 GagliardoNirenberg inequality. This technique has a wide application, especially for equations that do not satisfy the maximum principle (cf.[20]). 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
Due to the singularity of equation (1), problem (1)(3) has no classical solutions in general. To state the definition of the weak solution, we let and firstly define the class of nonnegative testing functions
Definition 1 A function is called a weak subsolution (supersolution) of problem (1)(3) in if the following conditions hold:
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.
Lemma 1Letk, αbe positive constants and. Ifis a nonnegative absolutely continuous function onsatisfying the problem
then we have the decay estimate
Proof
By solving the initial problem
and using the comparison principle, one can easily obtain the result. □
Lemma 2 (The GagliardoNirenberg inequality) [23]
Suppose that, , , and. We then have the inequality
whereCis a constant depending onN, m, r, j, k, andqsuch thatand. 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 1Suppose thatand. Then the nonnegative nontrivial weak solution of problem (1)(3) vanishes in finite time for any nonnegative initial data provided that eitherorλis sufficiently small.
Proof We first consider the case that . Multiplying both sides of (1) by u and integrating the result over Ω, we have
By Hölder’s inequality, we get the inequality
In particular, if , we get the inequality
from the two expressions above. By using the Sobolev embedding inequality, one can show that there exists an embedding constant such that
i.e.,
In particular, if , then the inequality above turns out to be
Since , we get , and hence, inequality (5) becomes
By Lemma 2, we get the inequality
where . Since , and , it can be easily seen that .
It then follows from (8) and Young’s inequality that
where and will be determined later. If we choose
By inequalities (7) and (10), we get the inequality
Here, we can choose and λ or small enough so that
By Lemma 1, we then obtain
where , which give the decay estimates in finite time for .
Secondly, we consider the case that . If , multiplying both sides of (1) by () and integrating the result over Ω, we get
By Hölder’s inequality, we have the inequality
In particular, if , we then get the inequality
by the two expressions above. By the Sobolev embedding inequality, one can see that there exists an embedding constant such that
Using Hölder’s inequality again, we have the inequality
From inequalities (11), (12), and (13), we then obtain the inequality
By Lemma 2, we can also have
where . Since and , one can easily see that . Then it follows from (15) and Young’s inequality that
where and will be determined later. If we choose
then and . We then have the inequality
by (16). From inequalities (14) and (17), we can also obtain the inequality
Here, we can choose and λ or small enough so that . Setting , we have from the inequality above. By Lemma 1, we obtain that
where , which give the decay estimates in finite time for such that .
If , one can show that there exists an embedding constant such that
by multiplying both sides of (1) by () and integrating the result over Ω, and the Sobolev embedding inequality. By using the inequality above and a similar argument as above, the following decay estimates can be obtained:
provided that
Theorem 2If, then the nonnegative nontrivial weak solution of problem (1)(3) vanishes in finite time provided that, orλis sufficiently small, and, where if, then, and if, then.
Proof Assume that . If , multiplying both sides of (1) by u and integrating the result over Ω, we have the equation
By (10) and (18), and using Hölder’s inequality, we get the inequality
Choosing small enough so that , we obtain the inequality
Hence, we have the inequality
provided that
and
where , from which and a similar argument as the one used in the proof of Theorem 1, the following decay estimates can be obtained:
If and , multiplying both sides of (1) by () and integrating the result over Ω, we get the equation
By (17) and (19), and using Hölder’s inequality, we obtain the inequality
Choosing small enough so that , we have
Therefore, we obtain the inequality
provided that
and
where , which yields the following decay estimates:
Since , we have , and hence, if , then .
Assume that . If is the first eigenvalue of the boundary problem
and , , is an eigenfunction corresponding to the eigenvalue , then for sufficiently small , it can be easily shown that is an upper solution of problem (1)(3) provided that , . We then have for by the comparison principle. Therefore, from equation (19), we can obtain the inequality
from which the following decay estimates can be obtained:
provided that
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 3Suppose thatand. Then the nonnegative nontrivial weak solution of problem (1)(3) vanishes in finite time for any nonnegative initial data provided thatβis sufficiently large.
Proof We first consider the case that . Multiplying both sides of (1) by u and integrating the result over Ω, and using Hölder’s inequality, we get
By Lemma 2, we have the inequality
where . Since , we have . It then follows from (20) and Young’s inequality that
where will be determined later. From (18) and (21), one can see that
We then obtain the inequality
by (6) and the inequality above, where . We can choose small enough so that . Once is fixed, we may choose β large enough so that
Hence, we have the inequality
from which the following decay estimates can be obtained by a similar argument as the one used in the proof of Theorem 1:
Secondly, we consider the case that . If , multiplying both sides of (1) by () and integrating the result over Ω, and then using Hölder’s inequality, we get
By Lemma 2, it can be shown that
where . Since , we have . It then follows from (22) and Young’s inequality that
where will be determined later. From (19) and (23), one can see that
By (12), (13), and the inequality above, we can obtain the inequality
where . We can choose small enough so that . Once is fixed, we can choose β large enough so that
Hence, we can obtain the inequality
from which the following decay estimates can be obtained:
Similarly, one can obtain the following decay estimates for :
Remark 3 One can see from Theorems 13 that the extinction of nonnegative nontrivial weak solutions to problem (1)(3) occurs when .
Remark 4 Theorems 13 all require , λ, or to be sufficiently small or β to be sufficiently large.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the manuscript and read and approved the final manuscript.
Acknowledgements
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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