Non-existence of local solutions of semilinear heat equations of Osgood type in bounded domains

Abstract

We establish a local non-existence result for the equation ut−Δu=f(u) with Dirichlet boundary conditions on a smooth bounded domain Ω⊂Rn and initial data in Lq(Ω) when the source term f is non-decreasing and limsups→∞s−γf(s)=∞ for some exponent γ>q(1+2/n). This allows us to construct a locally Lipschitz f satisfying the Osgood condition ∫1∞1/f(s)ds=∞, which ensures global existence for initial data in L∞(Ω), such that for every q with 1≤q<∞ there is a non-negative initial condition u0∈Lq(Ω) for which the corresponding semilinear problem has no local-in-time solution (‘immediate blow-up’).