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: (ProQuest: Formulae and/or non-USASCII text omitted), where [Omega] stands for a regular bounded domain of R super(N), Delta sub(p)u is the p-Laplacian defined by Delta sub(p)u = div (|[white triangle down]u| super(p-2)|[white triangle down]u), 2 < or = p < [infinity], delta > 0 and T > 0. The nonlinear term [functionof] : [Omega] x R x R super(N) arrow right R is a Caratheodory function satisfying the growth condition [functionof](x, s, [xi]) < or = (as super(q-1) + b) + c[xi] super(p-p/q) for a.a. x [setmembership] [Omega], s [setmembership] R sub(+) and |[xi]| > or = M where a, c, M > 0 and b > or = 0 are some constants and q [setmembership] [p, p*) where p* = pN/N-p if p < N and p* = [infinity] if p > N. We prove for any initial nonnegative data u sub(0) [setmembership] L super(r)([Omega]) with r > or = 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.