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

• Published in: Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 2010-08, Vol.140 (4), p.821-831
• Main Author: Mizoguchi, Noriko
• Format: Article
• Language: English
• Subjects: Blowing; Cauchy problem; Exponents; Heat equations; Mathematical problems; Partial differential equations; Self-similarity; Uniqueness
• ISSN: 0308-2105; 1473-7124
• DOI: 10.1017/S0308210509000444

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. Analysis257 (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.