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
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 (
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
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
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
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
Definition 1 A function
a.
b.
c. For 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.
Lemma 1Letk, αbe positive constants and
then we have the decay estimate
where
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
whereCis a constant depending onN, m, r, j, k, andqsuch that
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 that
Proof We first consider the case that
By Hölder’s inequality, we get the inequality
In particular, if
from the two expressions above. By using the Sobolev embedding inequality, one can
show that there exists an embedding constant
i.e.,
In particular, if
Since
By Lemma 2, we get the inequality
where
It then follows from (8) and Young’s inequality that
where
then
By inequalities (7) and (10), we get the inequality
Here, we can choose
Setting
By Lemma 1, we then obtain
where
Secondly, we consider the case that
By Hölder’s inequality, we have the inequality
In particular, if
by the two expressions above. By the Sobolev embedding inequality, one can see that
there exists an embedding constant
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
where
then
by (16). From inequalities (14) and (17), we can also obtain the inequality
Here, we can choose
where
If
by multiplying both sides of (1) by
provided that
where
Theorem 2If
Proof Assume that
By (10) and (18), and using Hölder’s inequality, we get the inequality
Choosing
Hence, we have the inequality
provided that
and
where
where
If
By (17) and (19), and using Hölder’s inequality, we obtain the inequality
Choosing
Therefore, we obtain the inequality
provided that
and
where
where
Since
Assume that
and
from which the following decay estimates can be obtained:
provided that
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
Theorem 3Suppose that
Proof We first consider the case that
By Lemma 2, we have the inequality
where
where
We then obtain the inequality
by (6) and the inequality above, where
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:
where
Secondly, we consider the case that
By Lemma 2, it can be shown that
where
where
By (12), (13), and the inequality above, we can obtain the inequality
where
Hence, we can obtain the inequality
from which the following decay estimates can be obtained:
where
Similarly, one can obtain the following decay estimates for
where
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
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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