Paper detail

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-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 $Ω_ε:=Ω\setminus B_ε(x_0)$ where $Ω$ is a smooth bounded domain in $\mathbb R^N$, $N\geq 3$ and $B_ε(x_0)$ is a ball of $\mathbb R^N$ of center $x_0\inΩ$ and radius $ε>0$ small. \\ We show that if $ε>0$ is small enough, then there exists a sign-changing stationary solution $ϕ_ε$ of \eqref{CH} such that the solution of \eqref{CH} with initial value $v_0=λϕ_ε$ blows up in finite time if $|λ-1|>0$ 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.