Infinite time blow-up for the three dimensional energy critical heat equation in bounded domains

Abstract

We consider the Dirichlet problem for the energy-critical heat equation \begin{equation*} \begin{cases} u_t=Δu+u^5,~&\mbox{ in } Ω\times \mathbb{R}^+,\\ u(x,t)=0,~&\mbox{ on } \partial Ω\times \mathbb{R}^+,\\ u(x,0)=u_0(x),~&\mbox{ in } Ω, \end{cases} \end{equation*} where $\Omega$ is a bounded smooth domain in ${\mathbb{R}}^3$. Let $H_{\gamma }(x,y)$ be the regular part of the Green function of $-\Delta -\gamma$ in $\Omega$, where $\gamma \in (0,{\lambda }_1)$ and ${\lambda }_1$ is the first Dirichlet eigenvalue of $-\Delta$. Then, given a point $q\in \Omega$ such that $3\gamma (q)<{\lambda }_1$, where $$\gamma (q)=\sup \{\gamma >0:H_{\gamma }(q,q)>0\},$$ we prove the existence of a non-radial global positive and smooth solution $u(x,t)$ which blows up in infinite time with spike in $q$. The solution has the asymptotic profile $$u(x,t)\sim 3^{\frac{1}{4}}(\frac{\mu (t)}{\mu (t{)}^2+|x-\xi (t){|}^2}{)}^{\frac{1}{2}}\text{as}t\to \infty ,$$ where $$-\ln \mu (t)=2\gamma (q)t(1+o(1)),\xi (t)=q+O(\mu (t))\text{as}t\to \infty .$$

arXiv comment74 pages, 1 figure