Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Table of Contents - 2010 - Volume 140, Issue 04  

Research Article

On backward self-similar blow-up solutions to a supercritical semilinear heat equation

Noriko Mizoguchia1

a1 Precursary Research for Embryonic Science and Technology (PRESTO), Japan Science and Technology Agency (JST), 4-1-8 Honcho Kawaguchi, Saitama 332-0012, Japan and Department of Mathematics, Tokyo Gakugei University, Koganei, Tokyo 184-8501, Japan (mizoguti@u-gakugei.ac.jp)

Abstract

We are concerned with a Cauchy problem for the semilinear heat equation

\begin{equation} \left. \begin{aligned} u_t&=\Delta u+u^p&&\text{in~}\mathbb{R}^N\times(0,T), \\[3pt] u(x,0)&=u_0(x)\geq0&&\text{in~}\mathbb{R}^N. \end{aligned} \right\} \label{ABSeqn}\tag{P} \end{equation} If $\smash{u(x,t)=(T-t)^{-1/(p-1)}\varphi((T-t)^{-1/2}x)}$ for $x\in\mathbb{R}^N$ and $t\in[0,T)$ with a solution $\varphi\not\equiv0$ of $$ \Delta\varphi-\tfrac{1}{2}y\nabla\varphi-\frac{1}{p-1}\varphi+\varphi^p=0\qts{in}\mathbb{R}^N, $$

then u is called a backward self-similar solution blowing up at t = T. Let pS and pL be the Sobolev and the Lepin exponents, respectively. It was shown by Mizoguchi (J. Funct. Analysis 257 (2009), 2911–2937) that k ≡ (p − 1)−1/(p−1) is a unique regular radial solution of (P) if p > pL. We prove that it remains valid for p = pL. We also show the uniqueness of singular radial solution of (P) for p > pS. These imply that the structure of radial backward self-similar blow-up solutions is quite simple for p ≥ pL.

(Received February 27 2009)

(Accepted September 24 2009)