Blow-up for a double nonlocal heat equation

TL;DR

This paper investigates the conditions under which solutions to a nonlocal heat equation blow up or exist globally, focusing on the influence of spatial kernel behavior and deriving the Fujita exponent for fractional operators.

Contribution

It provides a comprehensive analysis of blow-up phenomena for a class of nonlocal heat equations with general kernels, including singular and compactly supported cases, and determines the Fujita exponent for related fractional operators.

Findings

Blow-up or global existence depends solely on the spatial kernel at infinity.

Established the Fujita exponent for fractional derivatives similar to Caputo and fractional Laplacian.

Developed estimates for the fundamental solution using a Duhamel formula in a general setting.

Abstract

We study the blow-up question for the diffusion equation involving a nonlocal derivative in time defined by convolution with a nonnegative and nonincreasing kernel, and a nonlocal operator in space driven by a nonnegative radial L\'evy kernel. We show that the existence of solutions that blow up in finite time or exist globally depends only on the behaviour of the spatial kernel at infinity. A main difficulty of the work stems from estimating the fundamental pair defining the solution through a Duhamel formula, due to the generality of the setting, which includes singular or not, at the origin, spatial kernels, that can be either positive or compactly supported. As a byproduct we obtain that the Fujita exponent for the fractional type operators similar to the Caputo fractional derivative and the fractional Laplacian.