  We establish non-existence results for the Cauchy problem of some semilinear
heat equations with non-negative initial data and locally Lipschitz,
nonnegative source term $f$. Global (in time) solutions of the scalar ODE $\dot
v=f(v)$ exist for $v(0)>0$ if and only if the Osgood-type condition
$\int_{1}^{\infty}\frac{\dee s}{f(s)} =\infty$ holds; by comparison this
ensures the existence of global classical solutions of $u_t=\Delta u+f(u)$ for
bounded initial data $u_0\in L^{\infty}(\R^n)$. It is natural to ask whether
the Osgood condition is sufficient to ensure that the problem still admits
global solutions if the initial data is in $L^q(\R^n)$ for some $1\le
q<\infty$. Here we answer this question in the negative, and in fact show that
there are initial conditions for which there exists no local solution in
$L^1_{\rm loc}(\R^n)$ for $t>0$.
