Existence of infinitely many singular self-similar solutions for the semilinear heat equation

We show the existence of infinitely many singular positive forward self-similar solutions for the semilinear heat equation ${\mathrm{\partial }}_{t}u=\mathrm{\Delta }u+u^p$ for $x\in {\mathbb{R}}^N$, $t>0$, where $N\ge 3$ and $N\mathrm{/}(N-2)\le p\le (N+2)\mathrm{/}(N-2)$. We also investigate the asymptotic behavior and the bound of the profiles of the solutions as $\mathrm{\mid }x\mathrm{\mid }\to 0$ and $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$, respectively. Furthermore, we show nonuniqueness of positive soutions of the Cauchy problem for the equation with $p=N\mathrm{/}(N-2)$ by using singular self-similar solutions.