Blow-up rates for a fractional heat equation

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