Around a singular solution of a nonlocal nonlinear heat equation

TL;DR

This paper investigates the existence, asymptotic behavior, and blowup conditions of solutions to a nonlocal nonlinear heat equation with small initial data, highlighting the role of singular solutions and critical spaces.

Contribution

It provides new results on global existence and asymptotics for a nonlocal nonlinear heat equation with small data, including conditions for blowup and comparison with singular solutions.

Findings

Global solutions exist for small initial data.

Asymptotic behavior of subcritical solutions is characterized.

Conditions for finite time blowup are identified.

Abstract

We study the existence of global-in-time solutions for a nonlinear heat equation with nonlocal diffusion, power nonlinearity and suitably small data (either compared pointwisely to the singular solution or in the norm of a critical Morrey space). Then, asymptotics of subcritical solutions is determined. These results are compared with conditions on the initial data leading to a finite time blowup.