Blow-up for a fully fractional heat equation

TL;DR

This paper investigates the blow-up behavior of solutions to a fully fractional heat equation involving a nonlocal operator, identifying critical exponents for global existence and blow-up rates.

Contribution

It characterizes the critical exponents for blow-up and global existence in a fractional heat equation with a nonlocal operator, extending classical results to fractional settings.

Findings

Global existence exponent p_0=1

Fujita exponent p_*=1+2σ/(N+2(1-σ))

Blow-up rate near T: (T-t)^{-σ/(p-1)}

Abstract

We study the existence and behaviour of blowing-up solutions to the fully fractional heat equation $$ \mathcal{M} u=u^p,\qquad x\in\mathbb{R}^N,\;0<t<T $$ with $p>0$, where $\mathcal{M}$ is a nonlocal operator given by a space-time kernel $M(x,t)=c_{N,\sigma}t^{-\frac N2-1-\sigma}e^{-\frac{|x|^2}{4t}}{1}_{\{t>0\}}$, $0<\sigma<1$. This operator coincides with the fractional power of the heat operator, $\mathcal{M}=(\partial_t-\Delta)^{\sigma}$ defined through semigroup theory. We characterize the global existence exponent $p_0=1$ and the Fujita exponent $p_*=1+\frac{2\sigma}{N+2(1-\sigma)}$, and study the rate at which the blowing-up solutions below $p_*$ tend to infinity, $\|u(\cdot,t)\|_\infty\sim (T-t)^{-\frac\sigma{p-1}}$.