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

Abstract

We establish non-existence results for the Cauchy problem of some semilinear heat equations with non-negative initial data and locally Lipschitz, non-negative source term f. Global (in time) solutions of the scalar ODE v˙=f(v) exist for v(0)>0 if and only if the Osgood-type condition ∫1∞dsf(s)=∞ holds; by comparison this ensures the existence of global classical solutions of ut=Δu+f(u) for bounded initial data u0∈L∞(Rn). 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 Lq(Rn) for some 1⩽q<∞. 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 Lloc1(Rn) for t>0.