Global existence and blow-up of solutions for p-Laplacian evolution equation with nonlinear memory term and nonlocal boundary condition

Zhong Bo Fang,Jianyun Zhang

doi:10.1186/1687-2770-2014-8

Abstract

In this paper, we deal with an initial boundary value problem for a p-Laplacian evolution equation with nonlinear memory term and inner absorption term subject to a weighted linear nonlocal boundary condition. We find the effects of a weighted function as regards determining blow-up of nonnegative solutions or not and establish the precise blow-up estimate for the linear diffusion case under some suitable conditions.

Highlights

1 Introduction In the past decades, there have been many works dealing with global existence and blowup properties of solutions for nonlinear parabolic equations, especially the initial boundary value problems with nonlocal terms in equations or boundary conditions, we refer to [ – ] and references therein
He studied the asymptotic behavior of solutions and found the influence of the weight function on the existence of global and blow-up solutions
Subject to a weighted nonlinear nonlocal boundary, u(x, t) = f (x, y)ul(y, t) dy, (x, t) ∈ ∂ × (, T), where p, q ≥. They gave the conditions of global existence and blow-up of solutions and the blow-up rate of solutions for q =, l = by establishing an auxiliary function

Summary

Introduction

There have been many works dealing with global existence and blowup properties of solutions for nonlinear parabolic equations, especially the initial boundary value problems with nonlocal terms in equations or boundary conditions, we refer to [ – ] and references therein. For the study of an initial boundary value problem for local parabolic equations with nonlocal boundary condition, we refer to [ – ]. T ut = u + uq up ds, (x, t) ∈ × ( , T), subject to a weighted nonlinear nonlocal boundary, u(x, t) = f (x, y)ul(y, t) dy, (x, t) ∈ ∂ × ( , T), where p, q ≥ They gave the conditions of global existence and blow-up of solutions and the blow-up rate of solutions for q = , l = by establishing an auxiliary function. There are some important phenomena formulated as parabolic equations which are coupled with weighted nonlocal boundary conditions in mathematical models, such as thermoelasticity theory In this case, the solution u(x, t) describes the entropy per volume of the materia (see [ , ]). + ξ δ v–ξ– (t – T ) ωδ– ω rxi rxj + ξ δv–ξ– (t – T ) ωδω rxi rxj + ξ δ (ξ + )v–ξ– (t – T ) (t – T ) ω δ– ω rxi rxj

Therefore we have

We need to show that for sufficiently small

For r

Combining the results with

Choosing u to be u