Abstract

We study the semilinear parabolic problem u(t) - Delta u = f(u)(p(x)) in Omega x (0,T) with zero Dirichlet condition on the boundary of a smooth bounded domain Omega subset of R-N. We assume that p is an element of C(Omega, & Ropf;(+)) is continuous satisfying 0 < p(-) = min(x is an element of Omega) p(x) <= p(x) <= max(x is an element of Omega)p(x) = p(+) < infinity, f is an element of C([0,+infinity)) is non-decreasing, and the initial data are in Lr(Omega), 1 <= r < infinity. We prove the existence, non-existence and uniqueness of solution in the space L-infinity((0,T), L (R)(Omega)). In our approach we use the supersolution method which allows us to conclude that this solution belongs to the space C([0,T], Lr(Omega)) boolean AND L-loc(infinity)((0,T), L-infinity(Omega)).