On the extinction problem for a p-Laplacian equation with a nonlinear gradient source

where R Ω N ⊂ , N 1 ≥ , is an open bounded domain with smooth boundary Ω ∂ , p 1 2 < < , the parameters μ, l and q are positive, and u L W Ω Ω p 0 0 ( ) ( ) ∈ ∩ ∞ is a nonzero nonnegative function. Nonlinear partial differential equations arise in various fields of science and have attracted the attention of many scholars (see [1–6]). Model (1.1) can be used to describe the combustion process in combustion theory where u x t , ( ) represents the temperature of the combustible substance (see [7]). Model (1.1) can also be used to describe the evolution of the population density of a kind of biological species under the effect of certain natural mechanism in population dynamics where u x t , ( ) is the density of the species (see [8]). In particular, model (1.1) with l 0 = is often referred to as a generalized viscous Hamilton-Jacobi equation. Model (1.1) with l 0 = is also related to the Kardar-Parisi-Zhang equation in the physical theory of growth and roughening of surfaces (see [9,10]). One of the particular features of problem (1.1) is that the equation is singular at the points where u 0 ∇ = . Hence, generally there is no classical solution and we introduce the definition of the weak solution for problem (1.1) as follows.


Introduction
The main aim of this paper is devoted to studying the extinction properties of the weak solutions for the following p-Laplacian equation is a nonzero nonnegative function. Nonlinear partial differential equations arise in various fields of science and have attracted the attention of many scholars (see [1][2][3][4][5][6]). Model (1.1) can be used to describe the combustion process in combustion theory where u x t , ( ) represents the temperature of the combustible substance (see [7]). Model (1.1) can also be used to describe the evolution of the population density of a kind of biological species under the effect of certain natural mechanism in population dynamics where u x t , ( ) is the density of the species (see [8]). In particular, model (1.1) with l 0 = is often referred to as a generalized viscous Hamilton-Jacobi equation. Model (1.1) with l 0 = is also related to the Kardar-Parisi-Zhang equation in the physical theory of growth and roughening of surfaces (see [9,10]).
One of the particular features of problem (1.1) is that the equation is singular at the points where u 0 ∇ = . Hence, generally there is no classical solution and we introduce the definition of the weak solution for problem (1.1) as follows. Definition 1.1. By a local weak solution to problem (1.1), we understand a function u C T L 0, ; Ω 1 ( ( ) ) ∈ for some T 0 > , which moreover satisfies the following assumptions: Problem (1.1) exhibits various interesting qualitative properties, such as blow-up (or gradient blow-up), extinction, dead-core, and quenching, which reflect natural phenomena, according to various conditions on the parameters p, μ, l, and q, the initial data u x 0 ( ) and the domain Ω (see [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]). Because of this, problem (1.1) has attracted the attention of many mathematicians in the last decade. The authors of [9,[26][27][28] showed the local existence result and established the comparison principle of the weak solutions. When μ 0 = , DiBenedetto [1] and Yuan et al. [29] proved that the necessary and sufficient condition for the extinction to occur is p 1 2 < < . The authors of [7,30,31] considered the extinction behaviors of the solutions for problem (1.1) with q 0 = and proved that the critical extinction exponent of the solution is l p 1 = − . For the case l 0 = and μ 1 = − , Iagar and Laurençot [32] concerned with the following Cauchy problem Based on comparison principle and gradient estimates of the solutions, they classify the behavior of the solutions for large time, obtaining either positivity as t → ∞ for q p N N 1 > − + , optimal decay estimates as Recently, under the restrictive condition N p > , Liu and Mu [33,34] considered problem (1.1) with l 0 = and proved that q p 1 = − is the critical extinction exponent of the nonnegative weak solution.
To the best of our knowledge, there is no result on the extinction behaviors of the solutions to problem (1.1). From a physical point of view, u u div is called a fast diffusion term, which may cause the extinction phenomenon of the weak solution; μu u l q | | ∇ with μ 0 > is called a nonlinear reaction term, which may prevent the extinction phenomenon of the weak solution. Motivated by the works above, our main attention will be focused on what role of the competition between the fast diffusion term and the coupled nonlinear hot source it plays in determining whether the extinction phenomenon occurs or not. Compared with some previous models which only have source term μu l or μ u q | | ∇ , the coupled reaction term μu u l q | | ∇ is more general, and it can reflect some complicated natural phenomena more exactly. Consequently, the coupled reaction term μu u l q | | ∇ will bring more challenges while dealing with the integral norm estimates and require more skills.
The main results of this article are stated as follows.
Theorem 1.2. Assume that l q 0 p p 2 < ≤ < + and p q l 1 − < + , then the nonnegative weak solution of problem (1.1) vanishes in finite time provided that u 0 is sufficiently small. Furthermore, one has Theorem 1.3. Assume that q l p 0 1 < + < − , then for any nonzero nonnegative initial data u 0 , problem (1.1) admits at least one non-extinction solution provided that μ is sufficiently large. (2) For any nonzero nonnegative initial data u 0 , problem (1.1) admits at least one non-extinction solution provided that μ is sufficiently large.
Remark 1.5. Theorems 1.2, 1.3, and 1.4 tell us that q l p 1 + = − is the critical extinction exponent of the weak solution of problem (1.1). On the other hand, Theorems 1.2, 1.3, and 1.4 generalize and extend the previous results in [7,30,31,33,34] to a more general case.

Proof of the main results
In this section, we will give the conditions on the occurrence of the extinction phenomenon by using integral norm estimate approach. Meanwhile, the proof of the non-extinction results will be given by using super-solution and sub-solution methods.
, it is easily seen that p s l q s s p q p s q q s s p q 0 1 1 Then by Young's inequality and Hölder's inequality, it holds that where ε s μ 0, Noting that q l p 1 + > − , one can take u 0 so small that Integrating both sides of (2.6) with respect to the time variable from 0 to t, one can conclude that On the other hand, Hölder's inequality and Sobolev embedding inequality lead us to the following estimate: where κ 2 is the embedding constant, depending only on p and N . Inserting (2.10) into (2.9), one has then from (2.11), it follows that where u x Λ Λ Λ d 0. Integrating both sides of (2.12) with respect to the time variable on t 0, ( ), one arrives at (III) N 1 = and p 1 2 < < . Since p 1 2 < < , there exists an embedding constant κ 3 such that With the help of (2.9) and (2.15), one observes Integrating (2.17), we arrive at the following inequality: The proof of Theorem 1.2 is complete. □ By virtue of It remains to prove that x t , then one can immediately claim that I 0 < , which suggests that x t , 1 ( )is a strict non-extinction weak subsolution of problem (1.1).
On the other hand, one can prove that x t K u x , m a x1 , is a non-extinction supersolution of problem (1.1), and x t x t , , . Therefore, by an iteration process, one can conclude that there is at least a non-extinction solution u x t , ( ) of problem (1.1), which satisfies x t u x t , ,