Blow-up phenomena for a nonlinear parabolic problem with p-Laplacian operator under nonlinear boundary condition

Xiufen Wang,Yimin Shi

doi:10.1186/s13661-016-0663-y

Abstract

In this paper, we study the blow-up phenomena for a positive solution of a nonlinear parabolic problem with p-Laplacian operator under a nonlinear boundary condition. The sufficient conditions which ensure that the blow-up does occur at finite time are presented by constructing some appropriate auxiliary functions and using first-order differential inequality technique. Moreover, a lower bound and an upper bound for the blow-up time are derived when blow-up happens.

Highlights

1 Introduction The mathematical investigation of the blow-up phenomena of a solution to nonlinear parabolic equations and systems has received a great deal of attention during the last few decades [ – ]
Gave a lower bound for the blow-up time under the above condition
A lower bound for the blow-up time was obtained

Summary

Introduction

The mathematical investigation of the blow-up phenomena of a solution to nonlinear parabolic equations and systems has received a great deal of attention during the last few decades [ – ]. The authors in [ , ] considered an initial-boundary value problem for parabolic equations of the form. O is a bounded domain in R , is the Laplace operator, ∇ is the gradient operator, ∂O is the boundary of O. They proved that problem ( ) blows up at finite time T∗ if. The relative result in [ ] was extended to the case with nonlinear boundary condition by Liu [ ]. Enache in [ ] considered a more complicated case, in which he investigated the following class of quasilinear initial-boundary value problems:

Here n is the unit outer normal vector of

We further prove that

Proof We first compute