Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Single-point blow-up for a doubly degenerate parabolic equation with nonlinear source

Chunlai Mua1 and Rong Zenga2

a1 College of Mathematics and Statistics, Chongqing University, Chongqing 401331, People's Republic of China

a2 College of Mathematics and Statistics, Chongqing University, Chongqing 401331, People's Republic of China (zengrong654321@yahoo.com.cn)

Abstract

This paper deals with the positive solution to the doubly degenerate equation

$$u_t-\mathrm{div}(|\nabla u^m|^\sigma\nabla u^m)=u^\beta\quad\text{for }x\in\mathbb{R}^N,~t>0,$$

where σ > 0, m > 1, β > m(1 + σ). We prove single-point blow-up for a large class of radial decreasing solutions. Furthermore, the upper and lower estimates of the blow-up solution near the single blow-up point are obtained.

(Received January 24 2010)

(Accepted August 25 2010)

(Online publication June 03 2011)