Blow-up of nonnegative solutions of an abstract semilinear heat equation with convex source

TL;DR

This paper establishes a sufficient condition for the blow-up of nonnegative solutions to a broad class of semilinear heat equations, highlighting the interplay between diffusion and reaction in various geometric settings.

Contribution

It provides a unified framework for blow-up criteria applicable to diverse spaces like manifolds, graphs, and metric measure spaces, extending previous results.

Findings

Derived a general blow-up condition for solutions in $L^p$ spaces.

Applied the criterion to Laplacians on various geometric structures.

Unified and extended existing blow-up results in a broad setting.

Abstract

We give a sufficient condition for non-existence of global nonnegative mild solutions of the Cauchy problem for the semilinear heat equation $u' = Lu + f(u)$ in $L^p(X,m)$ for $p \in [1,\infty)$, where $(X,m)$ is a $\sigma$-finite measure space, $L$ is the infinitesimal generator of a sub-Markovian strongly continuous semigroup of bounded linear operators in $L^p(X,m)$, and $f$ is a strictly increasing, convex, continuous function on $[0,\infty)$ with $f(0) = 0$ and $\int_1^\infty 1/f < \infty$. Since we make no further assumptions on the behaviour of the diffusion, our main result can be seen as being about the competition between the diffusion represented by $L$ and the reaction represented by $f$ in a general setting. We apply our result to Laplacians on manifolds, graphs, and, more generally, metric measure spaces with a heat kernel. In the process, we recover and extend some older as well as recent results in a unified framework.