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Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void

  • Dengming Liu EMAIL logo and Qi Chen
Published/Copyright: September 24, 2024
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Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

This article is devoted to the global existence and extinction behavior of the weak solution to a fast diffusion p -Laplace equation with logarithmic nonlinearity and special medium void. By applying energy estimates approach, Hardy-Littlewood-Sobolev inequality, and some ordinary differential inequalities, the global existence result is proved and the sufficient conditions on the occurrence of the extinction and nonextinction phenomena are given.

Keywords: fast diffusion p-Laplace equation; global existence; extinction; nonextinction; logarithmic nonlinearity
MSC 2010: 35K20; 35K55

1 Introduction

Let Ω ⊂ R N ( N ≥ 2 ) be an open bounded domain containing the coordinate origin x = 0 , with smooth boundary ∂ Ω , ∣ x ∣ = ∑ i = 1 N x i 2 for x = ( x 1 , … , x N ) ∈ Ω . In this article, we take an interest in the global existence and extinction behavior of the solutions for a fast diffusion p -Laplace equation with logarithmic nonlinearity and special medium void

(1.1) ∣ x ∣ − s u t − div ( ∣ ∇ u ∣ p − 2 ∇ u ) = f ( u ) , ( x , t ) ∈ Ω × ( 0 , + ∞ ) , u ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × ( 0 , + ∞ ) , u ( x , 0 ) = u 0 ( x ) , x ∈ Ω ,

with

f ( u ) = ∣ u ∣ q − 2 u log ∣ u ∣ , if u ≠ 0 ; 0 , if u = 0 ,

where p , q ∈ ( 1 , 2 ) , and s ≥ 0 are three parameters, and the initial datum u 0 ( x ) ∈ L ∞ ( Ω ) ∩ W 0 1 , p ( Ω ) is nontrivial. By the assumption on the parameter q , one knows that lim ∣ u ∣ → 0 + f ( u ) = 0 , and hence, f ( u ) ∈ C ( R ) . From a physical point of view, problem (1.1) can be used to describe the flow of a compressible fluid in a homogeneous isotropic rigid porous medium. In this content, u ( x , t ) is the density of the fluid, ∣ x ∣ − s acts as the volumetric moisture content, ∣ ∇ u ∣ p − 2 ∇ u presents the momentum velocity, and f ( u ) denotes the source. The readers can refer to [1] for more details of the background of problem (1.1).

In the recent years, mathematicians have focused their attention on the parabolic models with logarithmic nonlinearities and have obtained lots of achievements [2–9]. In particular, Chen et al. [10] dealt with problem (1.1) with s = 0 and p = q = 2 . Under some suitable hypotheses on the initial datum, they proved the existence of the uniformly bounded global solution and the infinite time blow-up phenomenon by utilizing the potential well method and logarithmic Sobolev inequality. Le and Le [11,12] studied the existence and nonexistence of the global weak solutions of problem (1.1) with s = 0 and p , q > 2 . Exactly speaking, they concluded that if p > q , then for any u 0 ( x ) ∈ W 0 1 , p ( Ω ) , problem (1.1) admits a global solution; if p ≤ q , then there exists a weak solution of problem (1.1), which is global provided that u 0 ( x ) belongs to some specific stable sets, and blows up in finite time provided that u 0 ( x ) belongs to some specific unstable sets. Compared with the case s = 0 , there are fewer results for problem (1.1) with s ≠ 0 . Deng and Zhou [13] considered problem (1.1) with s ∈ [ 0 , 2 ) and p = q = 2 . They gave some threshold results on the global existence or infinite time blow-up of the solutions in the cases of subcritical and critical initial energies. Liao and Tan [14] generalized the results in [13] to the case 2 ≤ p < q < N p N − p . Unlike the semilinear case (i.e., p = 2 ) in which the blow-up solutions blow up at t = + ∞ , Liao and Tan [14] pointed out that the blow-up solutions will blow up in finite time when p > 2 , and got the upper and lower bounds of the blow-up time.

We say that the solution u ( x , t ) of problem (1.1) possesses finite time extinction property (i.e., u ( x , t ) vanishes in finite time) if there exists a T ∈ ( 0 , + ∞ ) such that u ( x , t ) is nontrivial for t ∈ ( 0 , T ) but u ( x , t ) ≡ 0 for t > T . As one of the most remarkable properties that distinguish nonlinear parabolic problems from the linear ones, finite time extinction can be used to explain certain natural phenomena, such as, the depletion of the reactants in a chemical reaction process, the extinction of the species in biopopulation dynamics, and so on. There exist a number of works on the extinction and nonextinction properties of the weak solutions to problem (1.1) in the case of power nonlinearities and its generalization [15–26], but there is a little known about the one with logarithmic nonlinearity. Very recently, Deng and Zhou [27] considered problem (1.1) with s = 0 and p , q ∈ ( 1 , 2 ) , by using integral norm estimates approach and some ordinary differential inequalities, they obtained the sufficient conditions on the extinction and nonextinction behaviors of the global weak solutions.

As far as we know, there are no results on the extinction and nonextinction of the global solutions to the parabolic model with both special medium void and logarithmic nonlinearity. How to evaluate the effects of the singular potential ∣ x ∣ − s and the logarithmic nonlinearity ∣ u ∣ q − 2 u log ∣ u ∣ on the occurrence of the extinction phenomenon of the global solution to problem (1.1)? The main goal of the present article is to give an answer to this question. The main results of this article are stated as follows.

Theorem 1.1

Assume that 1 < p < 2 , 1 < q < 2 , s ≥ 0 , and u 0 ( x ) ∈ L ∞ ( Ω ) ∩ W 0 1 , p ( Ω ) . Then the weak solution u ( x , t ) of problem (1.1) exists globally.

Theorem 1.2

Assume that max { 1 , s } < p < q < 2 . If the initial data u 0 ( x ) satisfies

(1.2) 2 max ∫ Ω ∣ x ∣ − s u 0 + p a + 2 q + β − p p a + 2 , ∫ Ω ∣ x ∣ − s u 0 − p a + 2 q + β − p p a + 2 ≤ C 4 C 5 − 1

with some β ∈ ( 0 , 2 − q ) , a ≥ max − 1 p , 2 ( N − p ) − p ( N − s ) p ( p − s ) . Then the weak solution u ( x , t )   of problem (1.1) vanishes in finite time, where C 4 and C 5 are two positive constants given by (3.12) and (3.13), respectively.

Theorem 1.3

Assume that 1 < q ≤ p < 2 and s ≥ 0 . If

∫ Ω ∣ x ∣ − s u 0 2 d x > 0 ,

and

(1.3) E ( u 0 ) ≤ 0 , when p = q ; E ( u 0 ) < 0 , when p > q .

Then the weak solution of problem (1.1) cannot vanish in finite time, where

E ( u 0 ) = 1 p ∫ Ω ∣ ∇ u 0 ∣ p d x − 1 q ∫ Ω ∣ u 0 ∣ q log ∣ u 0 ∣ d x + 1 q 2 ∫ Ω ∣ u 0 ∣ q d x .

The remainder of this article is organized as follows. In Section 2, some definitions and notations are given, and some useful auxiliary lemmas are collected. In Section 3, the global existence result is proved and the conditions on the occurrence of the extinction and nonextinction behaviors are discussed. The proofs of Theorems 1.1, 1.2, and 1.3 will be given in Section 3. Finally, a conclusion of this article and the future scope of our work are provided in Section 4.

2 Preliminaries

In this section, we first introduce some notations and definitions. For a given function f , we define its positive part f + and negative part f − as the forms

f + = max { f , 0 } , f − = max { − f , 0 } .

Then f = f + − f − and ∣ f ∣ = f + + f − . For given p ∈ [ 1 , + ∞ ) , we introduce the Hilbert space

W 1 , p ( Ω ) = u ∈ L p ( Ω ) : ∂ u ∂ x i ∈ L p ( Ω ) , i = 1 , 2 , … , N ,

endowed with the norm

∥ u ∥ W 1 , p ( Ω ) = ∥ u ∥ L p ( Ω ) p + ∥ ∇ u ∥ L p ( Ω ) p p .

Denote

W 0 1 , p ( Ω ) = { u ∈ W 1 , p ( Ω ) : u ∣ ∂ Ω = 0 } .

Due to Poincaré’s inequality, one can know that

∥ ∇ u ∥ L p ( Ω ) = ∫ Ω ∣ ∇ u ∣ p d x 1 p

is an equivalent norm to ∥ u ∥ W 1 , p ( Ω ) in W 0 1 , p ( Ω ) .

It is worth noting that the first equation in problem (1.1) is singular at the points where ∣ ∇ u ∣ = 0 when 1 < p < 2 , and hence, there is no classical solution in general. Throughout this article, we work with the weak solution of problem (1.1) in the sense of the following definition.

Definition 2.1

Let T > 0 . A function u = u ( x , t ) defined in Ω × ( 0 , T ) is called a weak solution of problem (1.1) if u ∈ C ( [ 0 , T ) ; W 0 1 , p ( Ω ) ) , ∣ x ∣ − s 2 u t ∈ L 2 ( 0 , T ; L 2 ( Ω ) ) , u ( x , 0 ) = u 0 ( x ) , and for any φ ∈ W 0 1 , p ( Ω ) , it holds that

(2.1) ∫ 0 T ∫ Ω ∣ x ∣ − s u t φ d x + ∫ 0 T ∫ Ω ∣ ∇ u ∣ p − 2 ∇ u ⋅ ∇ φ d x = ∫ 0 T ∫ Ω ∣ u ∣ q − 2 u log ∣ u ∣ φ d x .

Similar to Theorem 2.3 of [14] (or Theorem 3.2 of [11]), the local existence of the weak solution of problem (1.1) can be guaranteed by the Galerkin approximation method.

Next, we collect some useful auxiliary lemmas, which play a key role in the later proofs of our main results.

Lemma 2.1

(Hardy-Littlewood-Sobolev inequality, see [28]) Assume that N ≥ 2 , 1 < μ < N , 0 ≤ ϑ ≤ μ and σ = μ ( N − ϑ ) N − μ . Then, there exists a positive constant κ = κ ( μ , ϑ , N ) such that

(2.2) ∫ Ω ∣ u ( x ) ∣ σ ∣ x ∣ ϑ d x ≤ κ ∫ Ω ∣ ∇ u ∣ μ d x N − ϑ N − μ

holds for any u ∈ W 0 1 , μ ( Ω ) , where Ω ⊂ R N is a bounded domain.

Lemma 2.2

(see [29]) Assume that 0 < k < p ≤ 1 . Let y ( t ) be the solution of the following inequality:

d y d t + C y k ≤ γ y p , t > 0 , y ( 0 ) = y 0 > 0 ,

with C > 0 and 0 < γ < C y 0 k − p 2 . Then, there are two positive constants η and ξ such that

0 ≤ y ( t ) ≤ ξ e − η t

holds for any t ≥ 0 .

Lemma 2.3

(see [30]) Assume that θ , δ , and χ are three positive constants. Let y ( t ) be a nonnegative absolutely continuous function satisfying

d y d t + δ y θ ( t ) ≥ χ , t > 0 .

Then, we have

y ( t ) ≥ min y ( 0 ) , χ δ 1 θ .

Lemma 2.4

(see [31]) Assume that σ and ρ are two positive constants. Then we have

Ψ σ log Ψ ≤ ( e ρ ) − 1 Ψ σ + ρ for a l l Ψ > 0

and

∣ Ψ σ log Ψ ∣ ≤ ( e σ ) − 1 for a l l 0 < Ψ < 1 .

3 Proofs of the main results

Proof of Theorem 1.1

Let u ( x , t ) ∈ C ( [ 0 , T ) ; W 0 1 , p ( Ω ) ) fulfilling ∣ x ∣ − s 2 u t ∈ L 2 ( 0 , T ; L 2 ( Ω ) ) and u ( x , 0 ) = u 0 ( x ) be a weak solution of problem (1.1). Since Ω is a bounded domain in R N , there is a ball B ( 0 , R ) of radius R = max x = ( x 1 , x 2 , … , x N ) ∈ Ω ¯ ∑ i = 1 N x i 2 1 2 centered at x = 0 such that Ω ⊆ B ( 0 , R ) . Recalling that q ∈ ( 1 , 2 ) , one can choose β ∈ ( 0 , 2 − q ) such that q + β ∈ ( 1 , 2 ) . For this selected β , by Lemma 2.4, one has

log ∣ u ∣ ≤ 1 e β ∣ u ∣ β .

Taking the test function φ = u ( x , t ) in (2.1), one has

(3.1) 1 2 d d t ∫ Ω ∣ x ∣ − s u 2 d x + ∫ Ω ∣ ∇ u ∣ p d x = ∫ Ω ∣ u ∣ q log ∣ u ∣ d x ≤ 1 e β ∫ Ω ∣ u ∣ q + β d x .

Making use of Hölder’s inequality, one obtains

(3.2) ∫ Ω ∣ u ∣ q + β d x ≤ ∫ Ω ∣ x ∣ s ( q + β ) 2 − q − β d x 2 − q − β 2 ∫ Ω ∣ x ∣ − s u 2 d x q + β 2 ≤ ∫ B ( 0 , R ) ∣ x ∣ s ( q + β ) 2 − q − β d x 2 − q − β 2 ∫ Ω ∣ x ∣ − s u 2 d x q + β 2 = ω N ( 2 − q − β ) 2 N + ( s − N ) ( q + β ) R 2 N + ( s − N ) ( q + β ) 2 − q − β 2 − q − β 2 ︸ C 0 ∫ Ω ∣ x ∣ − s u 2 d x q + β 2 ,

where ω N is the surface area of the unit sphere ∂ B ( 0 , 1 ) . Combining (3.1) and (3.2), one arrives at

d d t ∫ Ω ∣ x ∣ − s u 2 d x ≤ 2 C 0 e β ∫ Ω ∣ x ∣ − s u 2 d x q + β 2 ,

which implies that

(3.3) ∫ Ω ∣ x ∣ − s u 2 d x ≤ C 0 ( 2 − q − β ) e β t + ∫ Ω ∣ x ∣ − s u 0 2 d x 2 − q − β 2 2 2 − q − β .

Substituting (3.3) into (3.2) suggests that

(3.4) ∫ Ω ∣ u ∣ q + β d x ≤ C 0 C 0 ( 2 − q − β ) e β t + ∫ Ω ∣ x ∣ − s u 0 2 d x 2 − q − β 2 q + β 2 − q − β .

Taking the test function φ = u t ( x , t ) in (2.1), and using Lemma 2.4, Cauchy’s inequality and Hölder’s inequality lead us to that

(3.5) ∫ Ω ∣ x ∣ − s ( u t ) 2 d x + 1 p d d t ∫ Ω ∣ ∇ u ∣ p d x = ∫ Ω ∣ u ∣ q − 2 u log ∣ u ∣ u t d x = ∫ Ω 1 = { x ∈ Ω : ∣ u ∣ ≥ 1 } ∣ u ∣ q − 2 u log ∣ u ∣ u t d x + ∫ Ω 2 = { x ∈ Ω : ∣ u ∣ < 1 } ∣ u ∣ q − 2 u log ∣ u ∣ u t d x ≤ 1 e β ∫ Ω 1 ∣ u ∣ q + β − 1 ∣ u t ∣ d x + 1 e ( q − 1 ) ∫ Ω 2 ∣ u t ∣ d x ≤ ε 1 e β + ε 2 e ( q − 1 ) ∫ Ω ∣ x ∣ − s ( u t ) 2 d x + 1 4 ε 2 e ( q − 1 ) ∫ Ω ∣ x ∣ s d x + 1 4 ε 1 e β ∫ Ω ∣ x ∣ s ( q + β ) 2 − q − β d x 2 − q − β q + β ∫ Ω ∣ u ∣ q + β d x 2 ( q + β − 1 ) q + β .

If ε 1 and ε 2 are sufficiently small such that ε 1 e β + ε 2 e ( q − 1 ) ≤ 1 . Then from (3.5), it follows that

(3.6) d d t ∫ Ω ∣ ∇ u ∣ p d x ≤ C 1 ∫ Ω ∣ u ∣ q + β d x 2 ( q + β − 1 ) q + β + C 2 ,

where

C 1 = p 4 ε 1 e β ∫ B ( 0 , R ) ∣ x ∣ s ( q + β ) 2 − q − β d x 2 − q − β q + β = p 4 ε 1 e β C 0 2 q + β

and

C 2 = p 4 ε 2 e ( q − 1 ) ∫ B ( 0 , R ) ∣ x ∣ s d x = p 4 ε 2 e ( q − 1 ) ⋅ ω N N + s R N + s .

Summing up (3.4) and (3.6), one immediately obtains

(3.7) d d t ∫ Ω ∣ ∇ u ∣ p d x ≤ C 1 C 0 2 ( q + β − 1 ) q + β C 0 ( 2 − q − β ) e β t + ∫ Ω ∣ x ∣ − s u 0 2 d x 2 − q − β 2 2 ( q + β − 1 ) 2 − q − β + C 2 .

Integrating (3.7) with respect to the time variable over the interval ( 0 , T ) yields that

∫ Ω ∣ ∇ u ∣ p d x ≤ e β C 1 q + β C 0 q + β − 2 q + β C 0 ( 2 − q − β ) e β T + ∫ Ω ∣ x ∣ − s u 0 2 d x 2 − q − β 2 2 ( q + β − 1 ) 2 − q − β + 1 + C 2 T + ∫ Ω ∣ ∇ u 0 ∣ p d x − e β C 1 q + β C 0 q + β − 2 q + β ∫ Ω ∣ x ∣ − s u 0 2 d x q + β 2 ,

which means that, for any T ∈ ( 0 , + ∞ ) , the W 1 , p norm of the weak solution u ( x , t ) is finite in [ 0 , T ) . That is to say, the weak solution u ( x , t ) of problem (1.1) is global in the sense of W 1 , p norm. The proof of Theorem 1.1 is complete.□

Proof of Theorem 1.2

Motivated by the method introduced by DiBenedetto [32], one can select, modulo a Steklov average, the test function φ = ∣ u ∣ p a u + in (2.1) to obtain

(3.8) 1 p a + 2 d d t ∫ Ω ∣ x ∣ − s u + p a + 2 d x + p a + 1 ( a + 1 ) p ∫ Ω ∣ ∇ u + a + 1 ∣ p d x = ∫ Ω ∣ u ∣ q − 2 u ∣ u ∣ p a u + log ∣ u ∣ d x ≤ 1 β ∫ Ω u + q + p a + β d x .

Recalling that max { 1 , s } < p < q < q + β < 2 and a ≥ max − 1 p , 2 ( N − p ) − p ( N − s ) p ( p − s ) , and by virtue of Hölder’s inequality and Hardy-Littlewood-Sobolev inequality, one has

∫ Ω ∣ x ∣ − s u + p a + 2 d x ≤ ∫ Ω ∣ x ∣ − s d x 1 − ( N − p ) ( p a + 2 ) p ( N − s ) ( a + 1 ) ∫ Ω ∣ x ∣ − s ( u + a + 1 ) p ( N − s ) N − p d x ( N − p ) ( p a + 2 ) p ( N − s ) ( a + 1 ) ≤ C 3 κ ( N − p ) ( p a + 2 ) p ( N − s ) ( a + 1 ) ∫ Ω ∣ ∇ u + a + 1 ∣ p d x p a + 2 p ( a + 1 ) ,

which means that

(3.9) C 3 − p ( a + 1 ) p a + 2 κ − N − p N − s ∫ Ω ∣ x ∣ − s u + p a + 2 d x p ( a + 1 ) p a + 2 ≤ ∫ Ω ∣ ∇ u + a + 1 ∣ p d x ,

where

C 3 = ∫ B ( 0 , R ) ∣ x ∣ − s d x 1 − ( N − p ) ( p a + 2 ) p ( N − s ) ( a + 1 ) = ω N N − s R N − s 1 − ( N − p ) ( p a + 2 ) p ( N − s ) ( a + 1 ) ,

and κ = κ ( p , s , N ) is the embedding constant given in Lemma 2.1. On the other hand, by applying Hölder’s inequality again, one obtains

(3.10) ∫ Ω u + q + p a + β d x ≤ ∫ Ω ∣ x ∣ s ( q + β + p a ) 2 − q − β d x 2 − q − β p a + 2 ∫ Ω ∣ x ∣ − s u + p a + 2 d x q + β + p a p a + 2 .

Letting

y ( t ) ≔ ∫ Ω ∣ x ∣ − s u + p a + 2 d x ,

and summing up (3.8), (3.9), and (3.10), one arrives at

(3.11) d d t y ( t ) + C 4 ( y ( t ) ) p ( a + 1 ) p a + 2 ≤ C 5 ( y ( t ) ) q + β + p a p a + 2 ,

where

(3.12) C 4 = ( p a + 1 ) ( p a + 2 ) ( a + 1 ) p C 3 − p ( a + 1 ) p a + 2 κ − N − p N − s

and

(3.13) C 5 = p a + 2 β ∫ B ( 0 , R ) ∣ x ∣ s ( q + β + p a ) 2 − q − β d x 2 − q − β p a + 2 = p a + 2 β ⋅ ω N ( 2 − q − β ) 2 N + s p a + ( s − N ) ( q + β ) R 2 N + s p a + ( s − N ) ( q + β ) 2 − q − β 2 − q − β p a + 2 .

Remembering that { 1 , s } < p < q < q + β < 2 and a ≥ max − 1 p , 2 ( N − p ) − p ( N − s ) p ( p − s ) , one can check that

0 < p ( a + 1 ) p a + 2 < p a + q + β p a + 2 < 1 .

And then, by our assumption (1.2) and Lemma 2.2, one can claim that there exist two positive constants ξ and η such that, for any t ≥ 0 ,

(3.14) 0 ≤ y ( t ) ≤ ξ e − η t .

Choosing

T 0 > max 0 , 1 η ln ξ 2 C 5 C 4 p a + 2 q + β − p ,

then it follows from (3.11) and (3.14) that

d y d t + C 4 2 y p ( a + 1 ) p a + 2 ≤ 0 , t ≥ T 0 .

Integrating both sides of the aforementioned inequality with respect to the time variable on [ T 0 , t ] , one has

0 ≤ y 2 − p p a + 2 ( t ) ≤ y 2 − p p a + 2 ( T 0 ) − C 4 ( 2 − p ) 2 ( p a + 2 ) ( t − T 0 ) ,

which suggests that there exists a

T 0 ′ ∈ T 0 , T 0 + 2 ( p a + 2 ) C 4 ( 2 − p ) y 2 − p p a + 2 ( T 0 )

such that

lim t → T 0 ′ − y ( t ) = lim t → T 0 ′ − ∫ Ω ∣ x ∣ − s u + p a + 2 ( t ) d x = 0 .

Moreover, one can conclude that

∫ Ω ∣ x ∣ − s u + ( t ) d x = ∫ Ω ∣ x ∣ − s u + p a + 2 ( t ) d x → 0

as t → T 0 ′ − for a = − 1 p , and

∫ Ω ∣ x ∣ − s u + ( t ) d x ≤ ∫ Ω ∣ x ∣ − s d x p a + 1 p a + 2 ∫ Ω ∣ x ∣ − s u + p a + 2 ( t ) d x 1 p a + 2 ≤ ω N N − s R N − s p a + 1 p a + 2 ∫ Ω ∣ x ∣ − s u + p a + 2 ( t ) d x 1 p a + 2 → 0

as t → T 0 ′ − for a > − 1 p .

On the other hand, by using the similar way, one can show that ∫ Ω ∣ x ∣ − s u − ( t ) d x will also vanish in finite time. Thus,

∫ Ω ∣ x ∣ − s ∣ u ( t ) ∣ d x = ∫ Ω ∣ x ∣ − s u + ( t ) d x + ∫ Ω ∣ x ∣ − s u − ( t ) d x

will vanish in finite time. The proof of Theorem 1.2 is complete.□

Proof of Theorem 1.3

Denote

E ( u ( t ) ) = 1 p ∫ Ω ∣ ∇ u ∣ p d x − 1 q ∫ Ω ∣ u ∣ q log ∣ u ∣ d x + 1 q 2 ∫ Ω ∣ u ∣ q d x .

A direct calculation shows that

d E ( u ( t ) ) d t = − ∫ Ω ∣ x ∣ − s ( u t ) 2 d x ,

which implies that

(3.15) E ( u ( t ) ) = E ( u 0 ) − ∫ 0 t ∫ Ω ∣ x ∣ − s ( u τ ) 2 d x d τ .

Set

M ( t ) = 1 2 ∫ Ω ∣ x ∣ − s u 2 d x .

Taking the derivative of M ( t ) with respect to t , and using (3.15) lead us to

(3.16) M ′ ( t ) = − p E ( u ( t ) ) + q − p q ∫ Ω ∣ u ∣ q log ∣ u ∣ d x + p q 2 ∫ Ω ∣ u ∣ q d x = − p E ( u 0 ) + ∫ 0 t ∫ Ω ∣ x ∣ − s ( u τ ) 2 d x d τ + q − p q ∫ Ω ∣ u ∣ q log ∣ u ∣ d x + p q 2 ∫ Ω ∣ u ∣ q d x ≥ − p E ( u 0 ) + q − p q ∫ Ω ∣ u ∣ q log ∣ u ∣ d x .

If p = q . Then from (3.16), one can immediately see that, for any t ≥ 0 ,

(3.17) M ( t ) ≥ M ( 0 ) − p E ( u 0 ) t .

Keeping in mind that

M ( 0 ) = ∫ Ω ∣ x ∣ − s u 0 2 d x > 0

and E ( u 0 ) ≤ 0 , then (3.17) gives us that M ( t ) > 0 , which means that the solution u ( x , t ) of problem (1.1) cannot vanish in finite time.

If p > q . Then by Lemma 2.4 and Hölder’s inequality, one has

(3.18) M ′ ( t ) ≥ − p E ( u 0 ) + q − p q ∫ Ω ∣ u ∣ q log ∣ u ∣ d x ≥ − p E ( u 0 ) − p − q e β q ∫ Ω ∣ u ∣ q + β d x ≥ − p E ( u 0 ) − C 0 ( p − q ) e β q ( M ( t ) ) q + β 2 .

Remembering that

M ( 0 ) = ∫ Ω ∣ x ∣ − s u 0 2 d x > 0

and

E ( u 0 ) < 0 ,

by combining (3.18) with Lemma 2.3, one can conclude that

M ( t ) ≥ min M ( 0 ) , − e β p q E ( u 0 ) C 0 ( p − q ) 2 q + β > 0 .

Thus, the solution u ( x , t ) of problem (1.1) cannot vanish in finite time. The proof of Theorem 1.3 is complete.□

4 Conclusions

In this article, we deal with the global existence and extinction behavior of the solution for a fast diffusion p -Laplace equation with logarithmic nonlinearity and special medium void. By analyzing the effect of the singular potential ∣ x ∣ − s and the logarithmic nonlinearity ∣ u ∣ q − 2 u log ∣ u ∣ on the global existence and extinction behaviors of the solution, along with the modified energy estimates approach, Hardy-Littlewood-Sobolev inequality and some ordinary differential inequalities, the global existence of the solution is proved and sufficient conditions on the occurrence of the extinction and nonextinction phenomena are obtained.

Our next work is to study the numerical extinction and nonextinction phenomena of the parabolic problems like (1.1). We hope to give some numerical examples and applications for our theoretical researches in the near future.

Acknowledgments

The authors are sincerely grateful to the anonymous reviewers for their careful reading, valuable comments, and constructive suggestions, which greatly improved the quality of this article.

  1. Funding information: This article was supported by Scientific Research Fund of Hunan Provincial Education Department (Grant No. 23A0361).

  2. Author contributions: All authors contributed significantly and equally to writing this article. All authors read and approved the final manuscript.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: Data are contained within the article.

References

[1] X. M. Deng and J. Zhou, Global existence, extinction, and non-extinction of solutions to a fast diffusion p-Laplace evolution equation with singular potential, J. Dyn. Control Syst. 26 (2020), no. 3, 509–523, DOI: https://doi.org/10.1007/s10883-019-09462-5. 10.1007/s10883-019-09462-5Search in Google Scholar

[2] Y. Cao and C. H. Liu, Initial boundary value problem for a mixed pseudo-parabolic p-Laplacian type equation with logarithmic nonlinearity, Electron. J. Differential Equations 2018 (2018), no. 116, 1–19. Search in Google Scholar

[3] H. Ding and J. Zhou, Global existence and blow-up for a mixed pseudo-parabolic p-Laplacian type equation with logarithmic nonlinearity, J. Math. Anal. Appl. 478 (2019), no. 2, 393–420, DOI: https://doi.org/10.1016/j.jmaa.2019.05.018. 10.1016/j.jmaa.2019.05.018Search in Google Scholar

[4] H. Ding and J. Zhou, Global existence and blow-up for a parabolic problem of Kirchhoff type with logarithmic nonlinearity, Appl. Math. Optim. 83 (2021), no. 11, 1651–1707, DOI: https://doi.org/10.1007/s00245-019-09603-z. 10.1007/s00245-019-09603-zSearch in Google Scholar

[5] Y. Z. Han, Blow-up at infinity of solutions to a semilinear heat equation with logarithmic nonlinearity, J. Math. Anal. Appl. 474 (2019), no. 1, 513–517, DOI: https://doi.org/10.1016/j.jmaa.2019.01.059. 10.1016/j.jmaa.2019.01.059Search in Google Scholar

[6] Y. J. He, H. H. Gao, and H. Wang, Blow-up and decay for a class of pseudo-parabolic p-Laplacian equation with logarithmic nonlinearity, Comput. Math. Appl. 75 (2018), no. 2, 459–469, DOI: https://doi.org/10.1016/j.camwa.2017.09.027. 10.1016/j.camwa.2017.09.027Search in Google Scholar

[7] H. J. Liu and Z. B. Fang, On a singular epitaxial thin-film growth equation involving logarithmic nonlinearity, Math. Methods Appl. Sci. 47 (2024), no. 7, 5699–5728, DOI: https://doi.org/10.1002/mma.9887. 10.1002/mma.9887Search in Google Scholar

[8] X. Z. Sun and B. C. Liu, Classification of initial energy to a pseudo-parabolic equation with p(x)-Laplacian, J. Dyn. Control Syst. 29 (2023), no. 3, 873–899, DOI: https://doi.org/10.1007/s10883-022-09629-7. 10.1007/s10883-022-09629-7Search in Google Scholar

[9] J. Zhou, Behavior of solutions to a fourth-order nonlinear parabolic equation with logarithmic nonlinearity, Appl. Math. Optim. 84 (2021), no. 1, 191–225, DOI: https://doi.org/10.1007/s00245-019-09642-6. 10.1007/s00245-019-09642-6Search in Google Scholar

[10] H. Chen, P. Luo, and G. W. Liu, Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity, J. Math. Anal. Appl. 422 (2015), no. 1, 84–98, DOI: https://doi.org/10.1016/j.jmaa.2014.08.030. 10.1016/j.jmaa.2014.08.030Search in Google Scholar

[11] C. N. Le and X. T. Le, Global solution and blow-up for a class of p-Laplacian evolution equations with logarithmic nonlinearity, Acta Appl. Math. 151 (2017), no. 1, 149–169, DOI: https://doi.org/10.1007/s10440-017-0106-5. 10.1007/s10440-017-0106-5Search in Google Scholar

[12] N. C. Le and T. X. Le, Existence and nonexistence of global solutions for doubly nonlinear diffusion equations with logarithmic nonlinearity, Electron. J. Qual. Theory Differ. Equ. 2018 (2018), no. 67, 1–25, DOI: https://doi.org/10.14232/ejqtde.2018.1.67. 10.14232/ejqtde.2018.1.67Search in Google Scholar

[13] X. M. Deng and J. Zhou, Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity, Commun. Pure Appl. Anal. 19 (2020), no. 2, 923–939, DOI: https://doi.org/10.3934/cpaa.2020042. 10.3934/cpaa.2020042Search in Google Scholar

[14] M. L. Liao and Z. Tan, Global existence and blow-up of solutions to a class of non-Newton filtration equations with singular potential and logarithmic nonlinearity, 2020, https://arxiv.org/pdf/2010.01483. Search in Google Scholar

[15] R. Ayazoglu, G. Alisoy, S. Akbulut, and T. A. Aydin, Existence and extinction of solutions for parabolic equations with nonstandard growth nonlinearity, Hacet. J. Math. Stat. 53 (2024), no. 2, 367–381, DOI: https://doi.org/10.15672/hujms.1106985. 10.15672/hujms.1106985Search in Google Scholar

[16] Y. G. Gu, Necessary and sufficient conditions of extinction of solution on parabolic equations, Acta Math. Sinica (Chinese Ser.) 37 (1994), no. 1, 73–79.Search in Google Scholar

[17] Y. Z. Han and X. Liu, Global existence and extinction of solutions to a fast diffusion p-Laplace equation with special medium void, Rocky Mountain J. Math. 51 (2021), no. 3, 869–881, DOI: https://doi.org/10.1216/rmj.2021.51.869. 10.1216/rmj.2021.51.869Search in Google Scholar

[18] C. H. Jin, J. X. Yin, and Y. Y. Ke, Critical extinction and blow-up exponents for fast diffusive polytropic filtration equation with sources, Proc. Edinb. Math. Soc. 52 (2009), no. 2, 419–444, DOI: https://doi.org/10.1017/S0013091507000399. 10.1017/S0013091507000399Search in Google Scholar

[19] D. M. Liu and C. Y. Liu, Global existence and extinction singularity for a fast diffusive polytropic filtration equation with variable coefficient, Math. Probl. Eng. 2021 (2021), 5577777, DOI: https://doi.org/10.1155/2021/5577777. 10.1155/2021/5577777Search in Google Scholar

[20] D. M. Liu and C. Y. Liu, On the global existence and extinction behavior for a polytropic filtration equation with variable coefficients, Electron. Res. Arch. 30 (2022), no. 2, 425–439, DOI: https://doi.org/10.3934/era.2022022. 10.3934/era.2022022Search in Google Scholar

[21] D. M. Liu and C. L. Mu, Critical extinction exponent for a doubly degenerate non-divergent parabolic equation with a gradient source, Appl. Anal. 97 (2018), no. 12, 2132–2141, DOI: https://doi.org/10.1080/00036811.2017.1359557. 10.1080/00036811.2017.1359557Search in Google Scholar

[22] D. M. Liu and M. J. Yu, On the extinction problem for a p-Laplacian equation with a nonlinear gradient source, Open Math. 19 (2021), no. 1, 1069–1080, DOI: https://doi.org/10.1515/math-2021-0091. 10.1515/math-2021-0091Search in Google Scholar

[23] X. Z. Sun, Z. Q. Han, and B. C. Liu, Classification of initial energy in a pseudo-parabolic equation with variable exponents and singular potential, Bull. Iranian Math. Soc. 50 (2024), no. 1, 10, DOI: https://doi.org/10.1007/s41980-023-00844-x. 10.1007/s41980-023-00844-xSearch in Google Scholar

[24] Y. Tian and C. L. Mu, Extinction and non-extinction for a p-Laplacian equation with nonlinear source, Nonlinear Anal. 69 (2008), no. 8, 2422–2431, DOI: https://doi.org/10.1016/j.na.2007.08.021. 10.1016/j.na.2007.08.021Search in Google Scholar

[25] J. Zhou and C. L. Mu, Critical blow-up and extinction exponents for non-Newton polytropic filtration equation with source, Bull. Korean Math. Soc. 46 (2009), no. 6, 1159–1173, DOI: https://doi.org/10.4134/BKMS.2009.46.6.1159. 10.4134/BKMS.2009.46.6.1159Search in Google Scholar

[26] J. X. Yin, J. Li, and C. H. Jin, Non-extinction and critical exponent for a polytropic filtration equation, Nonlinear Anal. 71 (2009), no. 1–2, 347–357, DOI: https://doi.org/10.1016/j.na.2008.10.082. 10.1016/j.na.2008.10.082Search in Google Scholar

[27] X. M. Deng and J. Zhou, Extinction and non-extinction of solutions to a fast diffusion p-Laplace equation with logarithmic non-linearity, J. Dyn. Control Syst. 28 (2022), no. 4, 759–769, DOI: https://doi.org/10.1007/s10883-021-09548-z. 10.1007/s10883-021-09548-zSearch in Google Scholar

[28] M. Badiale and G. Tarantello, A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics, Arch. Ration. Mech. Anal. 163 (2002), no. 4, 259–293, DOI: https://doi.org/10.1007/s002050200201. 10.1007/s002050200201Search in Google Scholar

[29] W. J. Liu and B. Wu, A note on extinction for fast diffusive p-Laplacian with sources, Math. Methods Appl. Sci. 31 (2008), no. 12, 1383–1386, DOI: https://doi.org/10.1002/mma.976. 10.1002/mma.976Search in Google Scholar

[30] B. Guo and W. J. Gao, Non-extinction of solutions to a fast diffusion p-Laplace equation with Neumann boundary conditions, J. Math. Anal. Appl. 422 (2015), no. 2, 1527–1531, DOI: https://doi.org/10.1016/j.jmaa.2014.09.006. 10.1016/j.jmaa.2014.09.006Search in Google Scholar

[31] M. L. Liao, The lifespan of solutions for a viscoelastic wave equation with a strong damping and logarithmic nonlinearity, Evol. Equ. Control Theory 11 (2022), no. 3, 781–792, DOI: https://doi.org/10.3934/eect.2021025. 10.3934/eect.2021025Search in Google Scholar

[32] E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. 10.1007/978-1-4612-0895-2Search in Google Scholar
Received: 2024-03-15
Revised: 2024-07-15
Accepted: 2024-08-30
Published Online: 2024-09-24

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2024-0064
Keywords for this article
fast diffusion p-Laplace equation; global existence; extinction; nonextinction; logarithmic nonlinearity
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  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
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