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

TL;DR

This paper constructs solutions to the three-dimensional energy-critical heat equation in bounded domains that blow up infinitely slowly at a specific point, revealing new blow-up behavior in such PDEs.

Contribution

It establishes the existence of non-radial global solutions with infinite-time blow-up in bounded domains for the critical heat equation, with detailed asymptotic profiles.

Findings

Solutions blow up at a point in infinite time.

Asymptotic profile of solutions is explicitly characterized.

Conditions on domain and parameters for blow-up are identified.

Abstract

We consider the Dirichlet problem for the energy-critical heat equation \begin{equation*} \begin{cases} u_t=\Delta u+u^5,~&\mbox{ in } \Omega \times \mathbb{R}^+,\\ u(x,t)=0,~&\mbox{ on } \partial \Omega \times \mathbb{R}^+,\\ u(x,0)=u_0(x),~&\mbox{ in } \Omega, \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 $$ u(x,t)\sim 3^{\frac{1}{4}} \bigg(\frac{\mu(t)}{\mu(t)^2+|x-\xi(t)|^2}\bigg)^{\frac{1}{2}} \quad \text{as}\quad t \to \infty, $$ where $$ -\ln \mu(t)= 2\gamma(q) t(1+o(1)),\quad \xi(t)=q+O\big(\mu(t)\big) \quad \text{as}\quad t \to \infty. $$