Upper estimates for blow-up solutions of a quasi-linear parabolic equation

Abstract

In this paper, we consider a quasi-linear parabolic equation u_t=u^p(x_{xx}+u). It is known that there exist blow-up solutions and some of them develop Type II singularity. However, only a few results are known about the precise behavior of Type II blow-up solutions for p>2. We investigated the blow-up solutions for the equation with periodic boundary conditions and derived upper estimates of the blow-up rates in the case of 2<p<3 and in the case of p=3, separately. In addition, we assert that if 2 le p le 3 then lim _{t nearrow T}(T-t)^{frac{1}{p}+varepsilon }max u(x,t)=0 z for any varepsilon >0 under some assumptions.