Local existence of solutions to some degenerate parabolic equation associated with the p-Laplacian

The existence of local (in time) solutions of the initial-boundary value problem for the following degenerate parabolic equation: u(t)(x, t) - Delta(p) u (x, t) - |u|(q-2)u (x, t) = f(x, t), (x, t) epsilon Omega x (0, T), where 2 <= p < q < +infinity, Omega is a bounded domain in R-N, f : Omega x (0, T) R is given and Delta(p) denotes the socalled p-Laplacian defined by Delta(p)u := del. (|del u|(p-2 del)u), with initial data u(0) epsilon L-r(Omega) is proved under r > N(q - p) / p without imposing any smallness on u(0) and f. To this end, the above problem is reduced into the Cauchy problem for an evolution equation governed by the difference of two subdifferential operators in a reflexive Banach space, and the theory of subdifferential operators and potential well method are employed to establish energy estimates. Particularly, L-r-estimates of solutions play a crucial role to construct a time-local solution and reveal the dependence of the time interval [0, T-0] in which the problem admits a solution. More precisely, T-0 depends only on |u(0)|L-r and f. (C) 2007 Elsevier Inc. All rights reserved.