A remark on self-similar solutions for a semilinear heat equation with critical Sobolev exponent

Abstract

The Cauchy problem for a semilinear heat equation $$ w_t = \Delta w + w^p\quad \text{in}\ \mathbf{R}^N \times (0,\infty) $$ with singular initial data $w(x,0) =\lambda a (x/|x|) |x|^{-2/(p-1)}$ for $x\in\mathbf{R}^N\setminus \{0\}$ is studied, where $N \gt 2$, $p=(N+2)/(N-2)$, $\lambda \gt 0$ is a parameter, and $a\ge0$, $a\not\equiv0$. We investigate the asymptotic properties of the profile of positive self-similar solutions to the problem as $\lambda\to0$ when $N = 3,4,5$.