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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 747613, 5 pages
Extinction and Nonextinction for the Fast Diffusion Equation
1College of Mathematics and Physics, Chongqing University, Chongqing 400044, China
2College of Elementary Education, Chongqing Normal University, Chongqing 400047, China
3School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, China
Received 28 February 2013; Accepted 28 April 2013
Academic Editor: Ru Dong Chen
Copyright © 2013 Chunlai Mu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper deals with the extinction and nonextinction properties of the fast diffusion equation of homogeneous Dirichlet boundary condition in a bounded domain of with . For , under appropriate hypotheses, we show that is the critical exponent of extinction for the weak solution. Furthermore, we prove that the solution either extinct or nonextinct in finite time depends strongly on the initial data and the first eigenvalue of with homogeneous Dirichlet boundary.
In this paper, we deal with the following fast diffusion equation with gradient absorption terms: where , , , with is an open bounded domain with smooth boundary, and is a nonzero positive function.
Equation (1) appears in a lot of applications to describe the evolution of diffusion processes, in particular, fast diffusion for . In combustion theory, for instance, the function represents the temperature, the term represents the thermal diffusion, and is a source.
Extinction and nonextinction are important properties for solutions of many evolutionary equations, especially for fast diffusion equations. In 1974, Kalashnikov  considered the Cauchy problem of equation and firstly introduced the definition of extinction for its solution; that is, there exists a finite time such that the solution is nontrivial for , but for all . In this case, is called the extinction time.
Since then, many authors became interested in the extinction and nonextinction of all kinds of evolutionary equations. For the following homogeneous Dirichlet boundary value problem: Gu  obtained that the nontrivial solutions of the problem (2) vanish identically in a finite time if and only if , which implies that strong absorption will cause extinction in a finite time. More results on the extinction for the problem (2) have also been obtained by many researchers, and we can refer to [3–7] and the references therein. Because of the occurrence of such a phenomenon for a diffusion equation with a different absorption term, that is, the absorption term is a nonnegative function of instead of being a nonnegative function of , Benachour et al. [8, 9] considered the following Cauchy problem for the viscous Hamilton-Jacobi equation: They proved that the nonnegative classical solutions to the problem (3) are extinct in finite time and have noncompact support if .
Generally, in the problems (2) and (3), there is a comparison between the diffusion term and the absorption term, and the absorption is sufficiently strong to lead any bounded nonnegative solution to zero in finite time. However, while both the source in problem (2) and the source in problem (3) are called the “cool source,” the nonlinear source in problem (1) is physically called the “hot source.” As far as we know, the type of “hot source” has complicated influences on the properties of solutions compared with the case of “cool source” . Thus, few works are concerned with extinction property for solutions to the evolutionary equations with “hot source.” In 2005, Li and Wu  gave some necessary and sufficient conditions of extinction for the solutions to the following problem with “hot source”: where . They proved that if , the solutions to the problem (4) with small initial data vanished in finite time and if , the maximal solution to the problem (4) is positive for all . Recently, Tian and Mu  studied the following -Laplacian equation with nonlinear “hot source”: They obtained that for , is the critical exponent of extinction for the weak solution to the problem (5). Also, they show that for and , the extinction and nonextinction of the solution to the problem (5) depend strongly on the first eigenvalue of the problem in , . For more results on the extinction properties of equations with “hot source,” we can refer to [12, 13] and the references therein.
By replacing the diffusion term with the gradient absorption terms in (4), in this paper, we devote to establish the conditions for the extinction of solution to the problem (1), which involves the “hot source” rather than the “cool source.”
In this section, we will give some definitions and lemmas which is useful to the proof of our results in the next section. For our convenience, we first define some sets as follows: It is well known that the problem (1) has no classical solution in general. We need to consider its weak solutions which is defined as follows.
Definition 1. For any , a function is called a weak solution to the problem (1) if the following equalities hold for any and :
Remark 2. Similarly, to define a subsolution (resp., supersolution) (resp., ) of the problem (1), we need only to set (resp., ) in , (resp., ) on , and the first equality in (7) is replaced by (resp., ) for every .
As a useful tool in the proof of our main results in the next section, a comparison principle of the following problem is needed: Under the following four hypotheses:(H1), , , and , s.t. and for all and ;(H2) and with for all ;(H3) with ;(H4) for any constant , if , in and , , then the functions and defined almost everywhere in by belong to and , respectively, we have the following lemma on the comparison principle of the problem (8).
Lemma 3 (see ). Assume that the hypotheses (H1)–(H4) hold. Let and be nonnegative solutions of (8) on with . Assume further that for any , there is a constant such that for all . Then, almost everywhere on .
3. Main Results
In this section, by applying the energy method introduced in  and the comparison result in Lemma 3 of the problem (8), we give the main results on the extinction and nonextinction of the solution to the problem (1).
3.1. Extinction of the Solution
Theorem 4. Assume that , and let be a weak solution of the problem (1). If , then, for sufficiently small initial data, there exists a finite time , such that for all .
Proof. First of all, multiplying the first equation of the problem (1) by and integrating over , we can obtain the following crucial equation to our proof:
Then, we prove the theorem by the following two cases.
First, we consider the case of . Let in (11), and we can get by the Hölder inequality and Poincare inequality that where is the Sobolev embedding constant depending only on and .
Since , , it follows from the Young’s inequality for any that Also, since the assumptions and imply that , we can obtain by the Hölder inequality that Therefore, from the previous inequalities (11)–(14), we can obtain the following differential inequality: Choose sufficiently small such that and initial data sufficiently small such that Thus, we can obtain that where .
By integrating the inequality (17), we can obtain that there exists a finite time such that which implies that vanishes in finite time .
Next, we consider the second case of . Let in (11), and we can get by the Poincare inequality that where is the Sobolev embedding constant depending only on , , and . Also, by the similar calculations as the proof in the first case, we can get from the Young’s inequality for any that Choose sufficiently small such that and initial data sufficiently small such that Then, it follows from the inequalities (11) and (19)–(21) that where − .
By integrating the inequality (22), we can obtain that there exists a finite time such that which implies that vanishes in finite time . This completes the proof of Theorem 4.
Theorem 5. If , the solution to the problem (1) vanishes in finite time for sufficiently small.
Proof. When , we note that the left sides of the inequalities (16) and (21) always equal 1. Thus, by a similar argument in the proof of Theorem 4, we can choose sufficiently small such that the inequalities (16) and (21) hold, which imply that there exist such that vanishes identically for all . This completes the proof of Theorem 5.
3.2. Nonextinction of the Solution
In this subsection, we investigate the conditions under which the solution of the problem (1) cannot become extinct.
Theorem 6. If , the weak solution of (1) cannot vanish in finite time for any nonnegative initial date with being sufficiently large.
Proof. In order to prove Theorem 6, we first define two useful functions as follows. The first function satisfying is the associated eigenfunction with the principal eigenvalue of the following problem: For , we define the second useful function as follows: where and . It follows from  that the function satisfies the following properties: Now, let be a function . Next, we will show that is a subsolution of the problem (1). In fact, from the definitions of functions , and the properties (26) of the function , we have that In order to prove that which implies that is a subsolution of the problem (1), we only show that Since and , we have that By choosing , we can get that which together with (29) implies that (28) holds. Therefore, is a subsolution of the problem (1). Moreover, since in and , we can obtain by the comparison principle that in , which implies that the weak solution of (1) cannot vanish in finite time. This completes the proof of Theorem 6.
This work is supported in part by NSF of China (11101069, 11071266) and in part by the Natural Science Foundation Project of CQ CSTC (2010BB9218).
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