Infinite time blow-up for the fractional heat equation with critical exponent

Abstract: We consider positive solutions for the fractional heat equation with critical exponent \begin{equation*} \begin{cases} u_t = -(-\Delta)^{s}u + u^{\frac{n+2s}{n-2s}}\text{ in } \Omega\times (0, \infty), u = 0\text{ on } (\mathbb{R}^n\setminus \Omega)\times (0, \infty), u(\cdot, 0) = u_0\text{ in }\mathbb{R}^n, \end{cases} \end{equation*} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^n$, $n > 4s$, $s\in (0, 1)$, $u:\mathbb{R}^n\times [0, \infty)\to \mathbb{R}$ and $u_0$ is a positive smooth initial datum with $u_0|_{\mathbb{R}^n\setminus \Omega} = 0$. We prove the existence of $u_0$ such that the solution blows up precisely at prescribed distinct points $q_1,\cdots, q_k$ in $\Omega$ as $t\to +\infty$. The main ingredient of the proofs is a new inner-outer gluing scheme for the fractional parabolic problems.