Existence versus instantaneous blow-up for linear heat equations with singular potentials

In this Note, we consider the linear heat equation u(t) - Delta u = a(x)u i n (0, T) x Omega, u = 0 on (0, T) x partial derivative Omega, and u(0) = u( 0) on Omega, where Omega subset of R-N is a smooth bounded domain. We assum e that a is an element of L-loc(1)(Omega), a greater than or equal to 0 and u(0) greater than or equal to 0. A simple condition on the potential a is necessary and sufficient for the existence of positive weak solutions that are global ill time and grow at most exponentially in time. We show that th is condition, based on the existence of a Hardy type inequality with weight a(x), is "almost" necessary for the local existence in time of positive we ak solutions. Applying these results to some "critical" potentials, we find new results on existence and on instantaneous and complete blow-lip of sol utions. (C) 1999 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.