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

摘要

We establish a local non-existence result for the equation ut?Δu =f(u) with Dirichlet boundary conditions on a smooth bounded domain Ω?R~n and initial data in L~q(Ω) when the source term fis non-decreasing and limsup_(s→∞)s~(?γ)f(s) =∞ for some exponent γ>q(1 +2). This allows us to construct a locally Lipschitz fsatisfying 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∈L~q(Ω) for which the corresponding semilinear problem has no local-in-time solution (‘immediate blow-up’).