Blow-up rates for a fractional heat equation

Abstract:

We study the speed at which nonglobal solutions to the fractional heat equation \begin{equation*} u_t+(-\Delta )^{\alpha /2} u=u^p, \end{equation*} with $0<\alpha <2$ and $p>1$, tend to infinity. We prove that, assuming either $p<p_F\equiv 1+\alpha /N$ or $u$ is strictly increasing in time, then for $t$ close to the blow-up time $T$ it holds that $\|u(\cdot ,t)\|_\infty \sim (T-t)^{-\frac 1{p-1}}$. The proofs use elementary tools, such as rescaling or comparison arguments.