article

Title: Self-similar solutions for a semilinear heat equation with critical Sobolev exponent

Authors: Yuki Naito

Issue: Volume 57 (2008), Issue 3, 1283-1316

Abstract:

$The Cauchy problem for a semilinear heat equation with singular initial data \[ \begin{cases} w_t = \Delta w + w^{p} & \mbox{in}\ \mathbb{R}^{N} \times (0,\infty)\\ w(x, 0) = \lambda a(x/|x|)|x|^{-2/(p-1)} &\mbox{in}\ \mathbb{R}^{N} \setminus \{0\} \end{cases} \] is studied, where $N > 2$, $p = (N+2)/(N-2)$, $\lambda > 0$ is a parameter, and $a \geq 0$, $a \not\equiv 0$. We show that there exists a constant $\lambda^{*} > 0$ such that the problem has at least two positive self-similar solutions for $\lambda \in (0, \lambda^{*})$ when $N = 3, 4, 5$, and that, when $N \geq 6$ and $a \equiv 1$, the problem has a unique positive radially symmetric self-similar solution for $\lambda \in (0, \lambda_{*})$ with some $\lambda_{*} \in (0, \lambda^{*})$. Our proofs are based on the variational methods and Pohozaev type arguments to the elliptic problem related to the profiles of self-similar solutions.$