Existence of a renormalized solutions to a nonlinear system in Orlicz spaces

Abstract

In this paper, we will be concerned with the existence of renormalized solutions to the following parabolic-elliptic system {?u ?t + Au = ?(u)|??|2 in QT = ? ? (0, T), ?div(?(u)??) = divF(u) in QT, u = 0 on ?? ? (0, T), ? = 0 on ?? ? (0, T), u(?, 0) = u0 in ?, where Au = ?div a(x, t, u,?u) is a Leray-Lions operator defined on the inhomogeneous Orlicz-Sobolev space W1,x 0 LM(QT) into its dual, M is a N-function related to the growth of a. M does not satisfy the ?2-condition, and ? and F are two Carath?odory functions defined in QT ? R.