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

作者：Laister Robert^*; Robinson James C; Sierzega Mikolaj

来源：Comptes Rendus Mathematique, 2014, 352(7-8): 621-626.

DOI：10.1016/j.crma.2014.05.010

We establish a local non-existence result for the equation ut - Delta u = f (u) with Dirichlet boundary conditions on a smooth bounded domain Omega subset of R-n and initial data in L-q(Omega) when the source term f is non-decreasing and lim sups, S f (s) = co for some exponent y > q(1 + 2/n). This allows us to construct a locally Lipschitz f satisfying the Osgood condition fr 1/f (s)ds =co, which ensures global existence for initial data in L (52), such that for every q with 1 < q < co there is a non-negative initial condition u0 E L5 (Q) for which the corresponding semilinear problem has no local-in-time solution ('immediate blow-up').

• 出版日期2014-7