Existence, uniqueness and asymptotic behavior of solutions for a singular parabolic equation

In this paper, we are concerned with a singular parabolic equation partial derivative v/partial derivative t - Delta v = f (x, t) - mu vertical bar del v vertical bar(2)/v in a smooth bounded domain Omega subset of R-N subject to zero Dirichlet boundary condition and initial condition phi >= 0. Under the assumptions on mu, phi and f (x, t), some existence and uniqueness results are obtained by applying parabolic regularization method and the sub-supersolutions method. We also discuss the asymptotic behaviors of solutions in the sense of L-2(0, T: W-0(1,2) (Omega)) and L-infinity(0, T; L-2 (Omega)) norms as mu -> 0 or mu -> infinity. As a byproduct we obtain the existence of solutions for some problems which blow up on the boundary. (C) 2009 Elsevier Inc. All rights reserved.