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We give a sufficient condition for non-existence of global nonnegative mildsolutions 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 $\sigma$-finitemeasure space, $L$ is the infinitesimal generator of a sub-Markovian stronglycontinuous semigroup of bounded linear operators in $L^p(X,m)$, and $f$ is astrictly increasing, convex, continuous function on $[0,\infty)$ with $f(0) =0$ and $\int_1^\infty 1/f < \infty$. Since we make no further assumptions onthe behaviour of the diffusion, our main result can be seen as being about thecompetition between the diffusion represented by $L$ and the reactionrepresented by $f$ in a general setting. We apply our result to Laplacians onmanifolds, graphs, and, more generally, metric measure spaces with a heatkernel. In the process, we recover and extend some older as well as recentresults in a unified framework.

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