Blow-up for sign-changing solutions of the critical heat equation in domains with a small hole

Abstract

We consider the critical heat equation \begin{equation} \label{CH}\tag{CH} \begin{array}{lr} v_t-Δv =|v|^{\frac{4}{n-2}}v & Ω_ε\times (0, +\infty) \\ v=0 & \partialΩ_ε\times (0, +\infty) \\ v=v_0 & \mbox{ in } Ω_ε\times \{t=0\} \end{array} \end{equation} in ${\Omega }_{\epsilon }:=\Omega \setminus B_{\epsilon }(x_0)$ where $\Omega$ is a smooth bounded domain in ${\mathbb{R}}^N$, $N\ge 3$ and $B_{\epsilon }(x_0)$ is a ball of ${\mathbb{R}}^N$ of center $x_0\in \Omega$ and radius $\epsilon >0$ small. \\ We show that if $\epsilon >0$ is small enough, then there exists a sign-changing stationary solution ${\varphi }_{\epsilon }$ of \eqref{CH} such that the solution of \eqref{CH} with initial value $v_0=\lambda {\varphi }_{\epsilon }$ blows up in finite time if $|\lambda -1|>0$ is sufficiently small.\\ This shows in particular that the set of the initial conditions for which the solution of \eqref{CH} is global and bounded is not star-shaped.