EXTINCTION AND NON-EXTINCTION OF SOLUTIONS TO A P-LAPLACE EQUATION WITH A NONLOCAL SOURCE AND AN ABSORPTION TERM

Yuzhu Han,Haixia Li,Wenjie Gao

doi:10.3846/13926292.2014.909896

Abstract

In this short paper, the authors investigate the extinction and non-extinction of solutions to a fast diffusive p-Laplace equation with a nonlocal source and an absorption term. By applying the super-solution and sub-solution methods, instead of energy estimate methods, they give a classification of the exponents and coefficients for the solutions to vanish in finite time or not, which generalizes some previous results.

Highlights

In this paper, we consider a fast diffusive p-Laplace equation with a nonlocal source and an absorption term of the following form ut = div |∇u|p−2∇u + a uq(y, t) dy − bur, Ω u(x, t) = 0, u(x, 0) = u0(x), x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0, x ∈ Ω, (1.1)where a, b, q, r > 0, 1 < p < 2, Ω is a bounded domain in RN (N ≥ 1) with smooth boundary ∂Ω, and the initial datum u0(x) is a nonnegative nontrivial function such that u0 ∈ L∞(Ω) ∩ W01,p(Ω).The equation in (1.1) is a p-Laplace equation perturbed by both a nonlinear nonlocal source term and an absorption term, which describes the fastY.Z
Problem (1.1) is a possible model for the diffusion system of some biological species with human-controlled distribution where u(x, t) represents the density of the species at position x and time t, div(|∇u|p−2∇u) portrays the mutation, −bur is the growth capacity of the species at location x and time t, whereas a Ω uq(y, t) dy denotes the human-controlled distribution
In a recent paper [4], Fang and Xu investigated the case r = 1, proved that q = p − 1 is the critical extinction exponent of solutions of problem (1.1) and gave the precise decay estimates, by using the methods of energy estimates. Their results show that the linear absorption term does not change the critical extinction exponents

Summary

Introduction

We consider a fast diffusive p-Laplace equation with a nonlocal source and an absorption term of the following form. In a recent paper [4], Fang and Xu investigated the case r = 1, proved that q = p − 1 is the critical extinction exponent of solutions of problem (1.1) and gave the precise decay estimates, by using the methods of energy estimates. Their results show that the linear absorption term does not change the critical extinction exponents.

Preliminaries and Main Results

Proof of the Main Results