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 = -(-Δ)^{s}u + u^{\frac{n+2s}{n-2s}}\text{ in } Ω\times (0, \infty), u = 0\text{ on } (\mathbb{R}^n\setminus Ω)\times (0, \infty), u(\cdot, 0) = u_0\text{ in }\mathbb{R}^n, \end{cases} \end{equation*} where $\hat{I}\text{c◯}$ 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 $u_0{|}_{{\mathbb{R}}^n\setminus \hat{I}\text{c◯}}=0$. We prove the existence of $u_0$ such that the solution blows up precisely at prescribed distinct points $q_1,\cdots ,q_k$ in $\hat{I}\text{c◯}$ as $t\to +\infty$. The main ingredient of the proofs is a new inner-outer gluing scheme for the fractional parabolic problems.