Non-existence of local solutions for semilinear heat equations of Osgood type

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 $\dot v=f(v)$existfor$v(0)>0$ifandonlyiftheOsgood-typecondition$\int_{1}^{\infty}\frac{\dee s}{f(s)} =\infty$ holds; by comparison this ensures the existence of global classical solutions of $u_t=\mathrm{\Delta }u+f(u)$ for bounded initial data $u_0\in L^{\mathrm{\infty }}({\mathbb{R}}^n)$. 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 $L^q({\mathbb{R}}^n)$ 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 $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1({\mathbb{R}}^n)$ for $t>0$.