Blow‐Up Analysis for a Quasilinear Degenerate Parabolic Equation with Strongly Nonlinear Source

Pan Zheng,Dengming Liu,Xianzhong Yao,Shouming Zhou,Chunlai Mu

doi:10.1155/2012/109546

Abstract

We investigate the blow‐up properties of the positive solution of the Cauchy problem for a quasilinear degenerate parabolic equation with strongly nonlinear source ut = div(|∇um|p−2∇ul) + uq, (x, t) ∈ RN × (0, T), where N ≥ 1, p > 2 , and m, l, q > 1, and give a secondary critical exponent on the decay asymptotic behavior of an initial value at infinity for the existence and nonexistence of global solutions of the Cauchy problem. Moreover, under some suitable conditions we prove single‐point blow‐up for a large class of radial decreasing solutions.

Highlights

In this paper, we consider the following Cauchy problem to a quasilinear degenerate parabolic equation with strongly nonlinear source ut div |∇um|p−2∇ul uq, x, t ∈ RN × 0, T, 1.1 u x, 0 u0 x, x ∈ RN, where N ≥ 1, p > 2, m, l, q > 1, and the initial data u0 x is nonnegative bounded and continuous.Equation 1.1 has been suggested as a mathematical model for a variety of physical problems see 1
A nonnegative measurable function u x, t defined in RN × 0, T is called a weak solution of the Cauchy problem 1.1 if for every bounded open set Ω with smooth boundary ∂Ω, u ∈ Cloc Ω × 0, T, um, ul ∈ Lploc 0, T ; W1,p Ω, and uφ dx
The first goal of this paper is to study the blow-up behavior of solution u x, t of 1.1 when the initial data u0 x has slow decay near x ∞

Summary

Introduction

We consider the following Cauchy problem to a quasilinear degenerate parabolic equation with strongly nonlinear source ut div |∇um|p−2∇ul uq, x, t ∈ RN × 0, T , 1.1 u x, 0 u0 x , x ∈ RN, where N ≥ 1, p > 2, m , l , q > 1, and the initial data u0 x is nonnegative bounded and continuous. Mu et al 20 investigated the secondary critical exponent for the doubly degenerate parabolic equation with slow decay initial values and obtained similar results. Some authors studied single-point blow-up for the Cauchy problem to nonlinear parabolic equations see 21, and the references therein by different methods. When p 2, l 1 and N 1, the Cauchy problem 1.1 has been investigated by Weissler in , and the author obtained that the solution blows up only at a single point. We consider single-point blow-up for a large number of radial decreasing solutions of the Cauchy problem 1.1 and give upper bound of the radial solution u r, t in a small neighborhood of the point x, t , where x 0, t T.

Blow-Up Case

Global Existence

Findings

Single Point Blow-Up