Existence of solutions to some degenerate parabolic equation associated with the p-Laplacian in the critical case

This paper is concerned with the initial-boundary value problem for the following degenerate parabolic equation: u(t) (x, t) - Delta(p)u(x, t) - vertical bar u vertical bar(q-2)u(x, t) = f (x, t) with initial data u(0) is an element of L-r (Omega). Akagi (2007) [1] established the existence of local (in time) solutions to this problem in the case r > N(q - p)/p; however, the critical case r = N(q - p)/p has been left as an open problem. In this paper, even in the critical case r = N(q - p)/p, the existence of solutions to the problem is established under a certain restriction on u0. The key to our proof is Tartar's inequality, which enables us to derive desired convergences of approximate solutions to the problem from the compactness of the embedding W-0(1,p) (Omega) subset of L-2 (Omega). Incidentally, any smoothness is not imposed on partial derivative Omega at all while a smooth boundary is needed in [1]. (C) 2013 Elsevier Ltd. All rights reserved.