ON SOME QUASILINEAR PARABOLIC EQUATIONS WITH NON-MONOTONE MULTIVALUED TERMS

In this paper, we study the existence of solutions to the initial-boundary value problem for the following parabolic differential inclusion: [GRAPHICS] . where Omega is a bounded open subset of R-N with smooth boundary partial derivative Omega, T > 0, Q(T) := [0, T] x Omega, Gamma(T) := [0, T] x partial derivative Omega, u(t) = partial derivative u/partial derivative t, Delta p is the p-Laplace differen-tial operator, partial derivative phi denotes the subdifferential of a proper lower semicontinuous convex function phi : R -> [0, infinity], and G : Q(T) x R -> 2(R)\{(sic)} is a nonmonotone multivalued mapping. The case where phi (sic) 0 and G(t, x, u) = |u|(q-2)u gives the prototype of our problem, denoted by (E)p. The existence of time-local strong solutions for (E)(p) is already studied by several authors. However, these results require a stronger assumption on q than that for the semi-linear case (E)(p) with p = 2. More precisely, it has been long conjectured that (E)(p) should admit a time-local strong solution for the Sobolev-subcritical range of q, i.e., for all q is an element of (2, p*) with p* = infinity for p >= N and p* = Np/N-p for p < N, which is the well-known fact for the semi-linear case (E)(p) with p = 2. bar right arrow The main purpose of the present paper is to show this conjecture holds true and to extend this classical study to the cases where u bar right arrow G(., ., u) is upper semicontinuous or lower semicontinuous, each one is a generalized notion of the continuity in the theory of multivalued analysis. We also discuss the extension of large or small local solutions along the lines of arguments developed in [28].