Bounded solutions to a quasilinear and singular parabolic equation with p-Laplacian

In this paper, we study the existence of a positive local in time solution for the following singular nonlinear problem with homogeneous Dirichlet boundary conditions: {partial derivative(t)u - Delta(p)u = u (delta) + f(x, u, del u) in (0, T) x Omega = Q(T,) u = 0 on (0, T) x partial derivative Omega, u > 0 in Q(T), u(0, x) = u(0) >= 0 in Omega, where Omega stands for a regular bounded domain of R-N, Delta(p)u is the p-Laplacian defined by Delta(p)u = div(vertical bar del u vertical bar(p-2)vertical bar u vertical bar), 2 <= p < infinity, delta > 0 and T > 0. The nonlinear term f : Omega x R x R-N -> R is a Caratheodory function satisfying the growth condition f(x, s, xi) <= (as(q-1) + b) + c vertical bar xi vertical bar(p-p/q) for a. a. x is an element of Omega, s is an element of R+ and vertical bar xi vertical bar >= M where a, c, M > 0 and b >= 0 are some constants and q is an element of [p, p*) where p* = pN/N-p if p < N and p* = infinity if p >= N. We prove for any initial nonnegative data u(0) is an element of L-r (Omega) with r >= 2 large enough, the existence of at least one weak solution to (P). In the case delta < 2 + 1/p-1, we prove the uniqueness of the solution and further regularity results. For that we use some estimates based on logarithmic Sobolev inequalities to get ultracontractivity of a closely related semi-group of linear operators. (C) 2014 Elsevier Ltd. All rights reserved.