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

We consider the Dirichlet problem for the energy-critical heat equation \(\begin{aligned} {\left\{ \begin{array}{ll} u_t=\Delta u+u^5 & \text {in} \quad \Omega \times \mathbb {R}^+,\\ u=0 & \text {on} \quad \partial \Omega \times \mathbb {R}^+,\\ u(x,0)=u_0(x) & \text {in} \quad \Omega , \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{u}_t=\mathrm{\Delta }u+u^5\hfill & \text{in}\mathrm{\Omega }\times {\mathbb{R}}^{+},\hfill \\ u=0\hfill & \text{on}\partial \mathrm{\Omega }\times {\mathbb{R}}^{+},\hfill \\ u(x,0)=u_0(x)\hfill & \text{in}\mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ where \(\Omega \) $\mathrm{\Omega }$ is a bounded smooth domain in \(\mathbb {R}^3\) ${\mathbb{R}}^3$ . Let \(H_\gamma (x,y)\) $H_{\gamma }(x,y)$ be the regular part of the Green function of \(-\Delta -\gamma \) $-\mathrm{\Delta }-\gamma$ in \(\Omega \) $\mathrm{\Omega }$ , where \(\gamma \in (0,\lambda _1)\) $\gamma \in (0,{\lambda }_1)$ and \(\lambda _1\) ${\lambda }_1$ is the first Dirichlet eigenvalue of \(-\Delta \) $-\mathrm{\Delta }$ . Then, given a point \(q\in \Omega \) $q\in \mathrm{\Omega }$ such that \(3\gamma (q)<\lambda _1\) $3\gamma (q)<{\lambda }_1$ , where \(\begin{aligned} \gamma (q) :=\sup \{ \gamma>0: H_\gamma (q,q)>0 \}, \end{aligned}\) $$\begin{array}{c}\hfill \gamma (q):=sup\{\gamma >0:H_{\gamma }(q,q)>0\},\end{array}$$ 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 0.1 \(\begin{aligned} u(x,t)\sim 3^{\frac{1}{4}} \left( \frac{\mu (t)}{\mu (t)^2+\vert x-\xi (t)\vert ^2}\right) ^{\frac{1}{2}} \quad \text {as}\quad t \rightarrow \infty , \end{aligned}\) $$\begin{array}{c}\hfill u(x,t)\sim 3^{\frac{1}{4}}{\left(\frac{\mu (t)}{\mu {(t)}^2+{|x-\xi (t)|}^2}\right)}^{\frac{1}{2}}\text{as}t\to \infty ,\end{array}$$ where \(\begin{aligned} -\ln (\mu (t))= 2\gamma (q) t(1+o(1)),\quad \xi (t)=q+O(\mu (t)) \quad \text {as}\quad t \rightarrow \infty . \end{aligned}\) $$\begin{array}{c}\hfill -ln(\mu (t))=2\gamma (q)t(1+o(1)),\xi (t)=q+O(\mu (t))\text{as}t\to \infty .\end{array}$$