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' = 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.