Existence and asymptotic behavior for a singular parabolic equation

Abstract

We prove global existence of nonnegative solutions to the singular parabolic equation ut - Deltau+chi ({u> 0})(- u(-beta) +lambdaf(u)) = 0 in a smooth bounded domain Omega subset of R-N with zero Dirichlet boundary condition and initial condition u(0) is an element of C(Omega), u(0) greater than or equal to 0. In some cases we are also able to treat u(0) is an element of L-infinity(Omega). Then we show that if the stationary problem admits no solution which is positive a. e., then the solutions of the parabolic problem must vanish in finite time, a phenomenon called "quenching". We also establish a converse of this fact and study the solutions with a positive initial condition that leads to uniqueness on an appropriate class of functions.