Extinction and decay estimates of solutions for a p-Laplacian evolution equation with nonlinear gradient source and absorption

Xianghui Xu,Zhong Bo Fang

doi:10.1186/1687-2770-2014-39

Abstract

We investigate the extinction properties of non-negative nontrivial weak solutions of the initial-boundary value problem for a p-Laplacian evolution equation with nonlinear gradient source and absorption terms.

Highlights

1 Introduction We are concerned with the initial-boundary value problem for a p-Laplacian evolution equation with gradient source and absorption terms, ut = div |∇u|p– ∇u + λ|∇u|r – βuq, x ∈, t >, ( . )
The p-Laplacian operator appears in the study of torsional creep
Motivated by the works mentioned above, and because there is little literature on the study of the extinction and nonextinction properties for parabolic equations with nonlinear gradient source and absorption terms, in this paper, our goal is to establish the sufficient conditions about the extinction and nonextinction of solutions for the problem ( . )

Summary

Introduction

Motivated by the works mentioned above, and because there is little literature on the study of the extinction and nonextinction properties for parabolic equations with nonlinear gradient source and absorption terms, in this paper, our goal is to establish the sufficient conditions about the extinction and nonextinction of solutions for the problem ) vanishes in finite time for any non-negative initial data u provided that λ is sufficiently small,and we have the following. ) vanishes in finite time for any non-negative initial data u provided that λ is sufficiently small, and u(·, t) ≤. ) vanishes in finite time for any non-negative initial data u provided that λ (or | |) is sufficiently small or β is sufficiently large (the detailed proof can be referred to [ ]).

Proof of Theorem 1

Proof of Theorem 2

Proof of Theorem 4

Proof of Theorem 5