Parabolic equations with nonlinear singularities

We show the existence of positive solutions u is an element of L(2)(0, T; H(0)(1)(Omega)) for nonlinear parabolic problems with singular lower order terms of the asymptote- type. More precisely, we shall consider both semilinear problems whose model is {ut - Delta u + u/1-u = f(x, t) in Omega x (0, T), u(x, 0) = u(0)(x) in Omega, u(x, t) = 0 on partial derivative Omega x (0, T), and quasilinear problems having natural growth with respect to the gradient, whose model is {u(t) - Delta u + |del u|(2)/u gamma = f (x, t) in Omega x (0, T), u(x, 0) = u(0)(x) in Omega, u(x, t) = 0 on partial derivative Omega x (0, T), with gamma > 0. Moreover, we prove a comparison principle and, as an application, we study the asymptotic behavior of the solution as t goes to infinity. (C) 2010 Elsevier Ltd. All rights reserved.