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

q + m . (2) 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 u(x, t) being the density of the fluid and α(x) � |x| s 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.


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: |x| − s u t − div ∇u m p− 2 ∇u m � u q , (x, t) ∈ Ω ×(0, +∞), u(x, t) � 0, (x, t) ∈zΩ ×(0, +∞), where Ω ⊂ R N (N > p) is a bounded domain with smooth boundary zΩ, x � (x 1 , . . . , x N ) ∈ Ω, |x| � ����������� x 2 1 + · · · + x 2 N , u 0 (x) is a nonnegative and bounded function with u m 0 ∈ W 1,p 0 (Ω), and the parameters m, s, p, and q satisfy 0 < m ≤ 1, 0 < m(p − 1) < 1, Inhomogeneous parabolic problems arise in a wide range of physical contexts (see for instance [1][2][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 u(x, t) being the density of the fluid and α(x) � |x| − s 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][5][6][7][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 u ∈ S � def u ∈ C(0, T; L 1 (Ω)), u ∈ L 2q (Ω × (0, T)) ∩ L 2 (Ω × (0, T)), ∇u m ∈ L p (Ω × (0, T))} for some T > 0, which moreover satisfies the following assumptions: (i) For any 0 ≤ ϕ ∈ S � def ϕ ∈ S, ϕ| zΩ � 0 and 0 < t 1 < t 2 < T, one has 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][10][11][12][13][14][15][16][17][18][19][20][21][22][23] and the references therein). For the case s � 0, the authors in [24][25][26] concerned with the global existence and blow-up properties of the solutions to problem (4) with m � 1 and p ≥ 2. Yuan et al. [27] considered problem (4) with m � 1 and f(u) � 0 and showed that the solution of problem (4) vanishes in finite time if and only if 1 < p < 2. Gu [28] studied problem (4) with m � 1 and f(u) � − u q and claimed that the necessary and sufficient condition on the occurrence of extinction phenomenon is p ∈ (1, 2) or q ∈ (0, 1). Tian and Mu [29] and Jin and Yin [30] studied problem (4) with m � 1 and f(u) � λu q with λ > 0 and showed that q � p − 1 is the critical extinction exponent of the solutions. When s � 0 and f(u) � λu q with λ > 0, Jin et al. [31] and Zhou and Mu [32] concluded that the critical extinction exponent of the solution to problem (1) is q � m(p − 1). Compared with s � 0, there are few literatures for the case s > 0. By Hardy inequality and potential well method, Tan [33] obtained the global existence and blow-up results of problem (4) with s � 2 and m � 1. Wang [34] generated the results in [33] to the case 0 < s ≤ 2. Zhou generated the results in [33] to the case s ≥ 2 and gave the global existence and blow-up results for (4) with p � 2 and m � 1 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 m � 1 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 Ω is a bounded domain in R N , then there is a ball B(0, R) ⊂ R N centered at 0 with radius such that Ω ⊆ B(0, R).
Let u(x, t) be a weak solution of problem (1). Define an energy functional as the following form: en, by (3), one can easily show that which tells us that E(u) is nonincreasing with respect to t. We state our main results as follows.

Theorem 1.
Suppose that the parameters m, p, q, and s satisfy (2), and the initial data u 0 (x) are a nonnegative and bounded function with u m 0 ∈ W 1,p 0 (Ω). Let u(x, t) be a solution of problem (1). en, the maximal existence time of where R, κ 1 , and κ 2 are given by (5), (12), and Lemma 2, respectively, then the solution u(x, t) of problem (1) vanishes in finite time.
then the solution u(x, t) of problem (1) does not possess extinction phenomenon. e rest of this article is organized as follows. In Section 2, we collect some useful auxiliary lemmas. e 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 eorem 1 will be given in Section 3.

Preliminaries
In this section, as preliminaries, we collect some well-known results, which play an important role in our proof of eorem 1.
Lemma 1 (see [37]). Suppose N > s and Ω ⊂ R N is a bounded domain. en, we have where B(0, R) is the ball in R N centered at 0 with radius and denotes the surface area of the unit sphere zB(0, 1), and Γ is the usual Gamma function.

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 u(x, t). (3), and using Hölder's inequality, one has which implies that Mathematical Problems in Engineering where Integrating (20) from 0 to t, one gets From (19) and (22), it follows that On the other hand, taking the test function ϕ � (u m ) t in (3), then by using Cauchy's inequality with ε and Hölder's inequality, one can obtain Let ε be sufficiently small to ensure that (4m/(m + 1) 2 ) − (2 mε/(m + 1)) ≥ 0, then by (21), (23), and (24), one has Integrating (25) from 0 to t yields that 4 Mathematical Problems in Engineering , (26) which means that the solution u(x, t) of the problem (1) is global. (3), then we can see that which tells us that On the other hand, taking the test function ϕ � (u m ) t in (3), then Cauchy's inequality with ε leads to Choosing ε ∈ (0,(2/(m + 1))) to guarantee that (4m/(m + 1) 2 ) − (2λmε/(m + 1)) ≥ 0, then by (29), one has which implies that en, the proof of the global existence result is complete. Now, we take our attention to the extinction singularity of the solution u(x, t) to problem (1). We denote α � p(N − s)/ (N − p). Noticing that 0 ≤ s < p, we can verify that α > p. Let a be a constant satisfying From (32), it follows that 0 < m(pa + 1) + 1 αm(a + 1) < 1.
Selecting the test function ϕ � u m(pa+1) (x, t) in (3), one has Making use of Hölder's inequality, one can find that
If 2C 2 < C 1 y (m(p− 1)− 1)/(m(pa+1)+1) (0), then Lemma 4 tells us that there are two positive constants η 1 and ξ 1 satisfying Putting T 0 � max 0, ((m(pa + 1) )}, then for any t > T 0 , (40) leads to which together with (39) yields Integrating above inequality from T 0 to t leads to 6 Mathematical Problems in Engineering e above inequality means that lim t⟶T 1 where If max ((1 − m)/2), m(p − 1) < q < 1. In view of Hölder's inequality, one has Combining (38) with (46), one can conclude that where Recalling that 0 < m(p − 1) < q < 1 and (32), one can check that en, by (47) and Lemma 4, one knows that there are two positive constants η 2 and ξ 2 satisfying provided that 2C 3 y (q− m(p− 1))/(m(pa+1)+1) (0) < C 1 . Setting T 2 � max 0, ((m(pa + 1) )}, then for any t > T 1 , (50) leads to which together with (47) yields e remainder proof is the same as the previous one in the case q � 1, and we omit it here. Up to now, the proof of the extinction phenomenon of the solution u(x, t) to problem (1) is complete. Now, we begin to prove the nonextinction result. Denoting and taking the derivative of M(t) with respect to t, one has If q � m(p − 1). From (8), one knows that E(u) is nonincreasing with respect to t. en, for any t ≥ 0, one has M ′ (t) ≥ − pE u 0 . (55) Integrating, one obtains where Since M(0) > 0 and E(u 0 ) < 0, then from (58) and Lemma 3, it follows that which means that the solution u(x, t) of problem (1) does not possess extinction phenomenon.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.