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

TL;DR

This paper studies the critical heat equation in domains with a small hole, demonstrating the existence of sign-changing stationary solutions that lead to finite-time blow-up for initial conditions close to these solutions.

Contribution

It constructs sign-changing stationary solutions in perforated domains and shows they induce finite-time blow-up for nearby initial data, revealing complex solution behaviors.

Findings

Existence of sign-changing stationary solutions in domains with small holes.

Finite-time blow-up occurs for initial data close to these stationary solutions.

The set of initial conditions leading to global solutions is not star-shaped.

Abstract

We consider the critical heat equation \begin{equation} \label{CH}\tag{CH} \begin{array}{lr} v_t-\Delta v =|v|^{\frac{4}{n-2}}v & \Omega_{\epsilon}\times (0, +\infty) \\ v=0 & \partial\Omega_{\epsilon}\times (0, +\infty) \\ v=v_0 & \mbox{ in } \Omega_{\epsilon}\times \{t=0\} \end{array} \end{equation} in $\Omega_{\epsilon}:=\Omega\setminus B_{\epsilon}(x_0)$ where $\Omega$ is a smooth bounded domain in $\mathbb R^N$, $N\geq 3$ 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 $\phi_{\epsilon}$ of \eqref{CH} such that the solution of \eqref{CH} with initial value $v_0=\lambda \phi_{\epsilon}$ 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.