Global Existence and Extinction Singularity for a Fast Diffusive Polytropic Filtration Equation with Variable Coefficient

Abstract

In this article, we deal with an inhomogeneous fast diffusive polytropic filtration equation. By using the energy estimate approach, Hardy–Littlewood–Sobolev inequality, and a series of ordinary differential inequalities, we prove the global existence result and obtain the conditions on the occurrence of the extinction phenomenon of the weak solution.

1. Introduction

Our main objectives in this article are to deal with the global existence and the extinction phenomenon of the inhomogeneous fast diffusive polytropic filtration equation:where is a bounded domain with smooth boundary , , , is a nonnegative and bounded function with , and the parameters , , , and satisfy

Inhomogeneous parabolic problems arise in a wide range of physical contexts (see for instance [1–3] and the references therein, where a more detailed physical background can be found). Problem (1) can be used to describe the compressible fluid flows in a homogeneous isotropic rigid porous medium with being the density of the fluid and acting as the volumetric moisture content. On the other hand parabolic models like (1), together with differential equation models, stochastic differential equations, and linear systems, are regarded as the powerful tools to solve lots of problems from control engineering, image processing, and other areas (see [4–8]). Because of the degeneracy and the singularity, problem (1) might not have classical solution in general, and hence, we introduce definition of the weak solution as follows.

Definition 1. By a local weak solution to problem (1), we understand a function for some , which moreover satisfies the following assumptions:(i) For any and , one has(ii) as with convergence in .

In the past few decades, many mathematicians have studied the global existence, blow-up, and extinction phenomena of the following parabolic equation:subject to various assumptions (see [9–23] and the references therein). For the case , the authors in [24–26] concerned with the global existence and blow-up properties of the solutions to problem (4) with and . Yuan et al. [27] considered problem (4) with and and showed that the solution of problem (4) vanishes in finite time if and only if . Gu [28] studied problem (4) with and and claimed that the necessary and sufficient condition on the occurrence of extinction phenomenon is or . Tian and Mu [29] and Jin and Yin [30] studied problem (4) with and with and showed that is the critical extinction exponent of the solutions. When and with , Jin et al. [31] and Zhou and Mu [32] concluded that the critical extinction exponent of the solution to problem (1) is . Compared with , there are few literatures for the case . By Hardy inequality and potential well method, Tan [33] obtained the global existence and blow-up results of problem (4) with and . Wang [34] generated the results in [33] to the case . Zhou generated the results in [33] to the case and gave the global existence and blow-up results for (4) with and in [35, 36], respectively. To the best knowledge of us, there is little work on the global existence and extinction behavior of problem (1). In a recent paper, Deng and Zhou [37] considered the special case and analysed the effect of the singular potential on the global existence and extinction behavior of the solutions.

In order to state well our results, we first introduce some definitions, fundamental facts, and useful symbols. Since is a bounded domain in , then there is a ball centered at 0 with radiussuch that .

We denote the norm of by and the norm of by ; that is, for any ,and for any , . According to Poincaré’s inequality, one can see that is equivalent to in , and hence, we equip with the norm .

Let be a weak solution of problem (1). Define an energy functional as the following form:

Then, by (3), one can easily show thatwhich tells us that is nonincreasing with respect to .

We state our main results as follows.

Theorem 1. Suppose that the parameters , , , and satisfy (2), and the initial data are a nonnegative and bounded function with . Let be a solution of problem (1). Then, the maximal existence time of is ; that is, is a global solution. Moreover,(i)If and and there is a constant with such thatwhere , , and are given by (5), (12), and Lemma 2, respectively, then the solution of problem (1) vanishes in finite time.(ii)If andwherethen the solution of problem (1) does not possess extinction phenomenon.

The rest of this article is organized as follows. In Section 2, we collect some useful auxiliary lemmas. The last section is mainly focused on the global existence and the conditions on the occurrence of the extinction phenomenon of the solution. By Hardy–Littlewood–Sobolev inequality and some ordinary differential inequalities, the proof of Theorem 1 will be given in Section 3.

2. Preliminaries

In this section, as preliminaries, we collect some well-known results, which play an important role in our proof of Theorem 1.

Lemma 1 (see [37]). Suppose and is a bounded domain. Then, we havewhere is the ball in centered at 0 with radius satisfying anddenotes the surface area of the unit sphere , and is the usual Gamma function.

Lemma 2 (see [38]). Suppose , , , and . Then, there is a positive constant such thatholds for any , where is a bounded domain.

Lemma 3 (see [39]). Assume , , and are positive constants. Let be a nonnegative absolutely continuous function satisfying

Then, we have

Lemma 4 (see [40]). Suppose . Let be the solution of the ordinary differential inequality:where and . Then, there are two positive constants and such that, for ,

3. Proof of Theorem 1

In this section, we will give the proof of the global existence result and the conditions on the occurrence of the extinction phenomenon of the solution .

Case 1. If . Taking the test function in (3), and using Hölder’s inequality, one haswhich implies thatwhereIntegrating (20) from 0 to , one getsFrom (19) and (22), it follows thatOn the other hand, taking the test function in (3), then by using Cauchy’s inequality with and Hölder’s inequality, one can obtainLet be sufficiently small to ensure that , then by (21), (23), and (24), one hasIntegrating (25) from 0 to yields thatwhich means that the solution of the problem (1) is global.

Case 2. If , taking the test function in (3), then we can see thatwhich tells us thatOn the other hand, taking the test function in (3), then Cauchy’s inequality with leads toChoosing to guarantee that , then by (29), one haswhich implies thatThen, the proof of the global existence result is complete.
Now, we take our attention to the extinction singularity of the solution to problem (1). We denote . Noticing that , we can verify that . Let be a constant satisfyingFrom (32), it follows thatSelecting the test function in (3), one hasMaking use of Hölder’s inequality, one can find thatwhere is the same as that in (12). Since and , one can deduce that . This together with the assumption one has . Meanwhile, recalling that , then it follows from Lemma 2 thatwhere is given in Lemma 2. Combining (35) with (32), one seesExploiting (34) and (37), one can arrive atIn what follows, for the sake of simplicity, we denote , and .
If , then from (38), one can immediately know thatwhere . Remembering that and (32), one can check that . If , then Lemma 4 tells us that there are two positive constants and satisfyingPutting , then for any , (40) leads towhich together with (39) yieldsIntegrating above inequality from to leads toThe above inequality means thatwhereIf . In view of Hölder’s inequality, one hasCombining (38) with (46), one can conclude thatwhereRecalling that and (32), one can check thatThen, by (47) and Lemma 4, one knows that there are two positive constants and satisfyingprovided that . Setting , then for any , (50) leads towhich together with (47) yieldsThe remainder proof is the same as the previous one in the case , and we omit it here. Up to now, the proof of the extinction phenomenon of the solution to problem (1) is complete.
Now, we begin to prove the nonextinction result. Denotingand taking the derivative of with respect to , one hasIf . From (8), one knows that is nonincreasing with respect to . Then, for any , one hasIntegrating, one obtainswhich tells us that since and ; that is, the solution of problem (1) does not possess extinction phenomenon.
If , with the help of Hölder’s inequality, one getsExploiting (54) and (57), one can claim thatwhereSince and , then from (58) and Lemma 3, it follows thatwhich means that the solution of problem (1) does not possess extinction phenomenon.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This paper was supported by the Natural Science Foundation of Hunan Province (Grant no. 2019JJ50160), Scientific Research Fund of Hunan Provincial Education Department (Grant no. 20A174), and Scientific Research Fund of Hunan University of Science and Technology (Grant no. KJ2123).