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

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 ${\mathrm{\Omega }}_{ϵ}:=\mathrm{\Omega }\setminus B_{ϵ}(x_0)$ where $\mathrm{\Omega }$ is a smooth bounded domain in ${\mathbb{R}}^N$, $N\ge 3$ and $B_{ϵ}(x_0)$ is a ball of ${\mathbb{R}}^N$ of center $x_0\in \mathrm{\Omega }$ and radius $ϵ>0$ small. \ We show that if $ϵ>0$ is small enough, then there exists a sign-changing stationary solution ${\varphi }_{ϵ}$ of \eqref{CH} such that the solution of \eqref{CH} with initial value $v_0=\lambda {\varphi }_{ϵ}$ blows up in finite time if $\mathrm{\mid }\lambda -1\mathrm{\mid }>0$ 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.