Existence of mild solutions for a singular parabolic equation and stabilization

In this paper, we study the existence and the uniqueness of a positive mild solution for the following singular nonlinear problem with homogeneous Dirichlet boundary conditions: {partial derivative(t)u - Delta(p)u = u(-delta) + f(x, u) in (0, T) x Omega (def)(=) 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 operator defined by Delta(p)u = div(vertical bar del u vertical bar(p-2)vertical bar del u vertical bar), 1 < p < infinity, delta > 0 and T > 0. The nonlinear term f : Omega x R -> R is a bounded below Caratheodory function and nonincreasing with respect to the second variable (for a.e. x epsilon Omega). We prove for any initial positive data u(0) epsilon <(D(A))over bar>(L infinity) the existence of a mild solution to (S-t). Then, we deduce some stabilization results for problem (S-t) in L-infinity(Omega) when p >= 2. This complements some results obtained in [2] stated with the additional restriction delta < 2 + 1/p-1.