  We give a simple proof of a lower bound for the Dirichlet heat kernel in
terms of the Gaussian heat kernel. Using this we establish a non-existence
result for semilinear heat equations with zero Dirichlet boundary conditions
and initial data in $L^q(\Omega)$ when the source term $f$ is non-decreasing
and $\limsup_{s\to\infty}s^{-\gamma}f(s)=\infty$ for some $\gamma>q(1+2/n)$.
This allows us to construct a locally Lipschitz $f$ satisfying the Osgood
condition $\int_{1}^{\infty}1/f(s)\ \,\d s =\infty$, which ensures global
existence for bounded initial data, such that for every $q$ with $1\le
q<\infty$ there is an initial condition $u_0\in L^q(\Om)$ for which the
corresponding semilinear problem has no local-in-time solution.
