  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.
