A Blow-up Dichotomy for Semilinear Fractional Heat Equations

TL;DR

This paper establishes a precise criterion distinguishing when positive solutions to fractional semilinear heat equations blow up in finite time, linking PDE behavior to an associated ODE.

Contribution

It introduces a blow-up dichotomy for fractional heat equations, providing a necessary and sufficient condition for finite-time blow-up in terms of the source term.

Findings

Derived a necessary and sufficient condition for blow-up

Linked PDE blow-up behavior to an associated ODE

Established a blow-up dichotomy for fractional heat equations

Abstract

We derive a blow-up dichotomy for positive solutions of fractional semilinear heat equations on the whole space. That is, within a certain class of convex source terms, we establish a necessary and sufficient condition on the source for all positive solutions to become unbounded in finite time. Moreover, we show that this condition is equivalent to blow-up of all positive solutions of a closely-related scalar ordinary differential equation.