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

Abstract

We give a sufficient condition for non-existence of global nonnegative mild solutions of the Cauchy problem for the semilinear heat equation $u^{\prime }=Lu+f(u)$ 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.

arXiv comment18 pages; replaced a reference in the introduction, rewrote some paragraphs (outside of proofs) to improve clarity, corrected typos