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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 $\dot v=f(v)$ 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 $u_t=Δu+f(u)$ 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 $t>0$.