On Fila-King Conjecture in Dimension Four

TL;DR

This paper constructs and proves the stability of an infinite time blow-up solution for the four-dimensional energy critical heat equation, confirming a conjecture by Fila and King about the solution's asymptotic behavior.

Contribution

It provides the first rigorous construction and stability analysis of infinite time blow-up solutions in the 4D energy critical heat equation, confirming a longstanding conjecture.

Findings

Existence of a positive infinite time blow-up solution with logarithmic growth.

Proof of stability of the constructed blow-up solution.

Confirmation of Fila and King's conjecture in four dimensions.

Abstract

We consider the following Cauchy problem for the four-dimensional energy critical heat equation \begin{equation*} \begin{cases} u_t=\Delta u+u^{3} ~&\mbox{ in }~ {\mathbb R}^4 \times (0,\infty),\\ u(x,0)=u_0(x) ~&\mbox{ in }~ {\mathbb R}^4. \end{cases} \end{equation*} We construct a positive infinite time blow-up solution $u(x,t)$ with the blow-up rate $ \| u(\cdot,t)\|_{L^\infty({\mathbb R}^4)} \sim \ln t$ as $t\to \infty$ and show the stability of the infinite time blow-up. This gives a rigorous proof of a conjecture by Fila and King \cite[Conjecture 1.1]{filaking12}.