On Fila-King Conjecture in Dimension Four

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}.