  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}.

Analysis of PDEs