Nonlinear Analysis 60 (2005) 1–35 www.elsevier.com/locate/na Strongly nonlinear parabolic equations with natural growth terms in Orlicz spaces A. Elmahi a , ∗ , D. Meskine b a C.P.R., BP 49, Fès, Morocco b Département de Mathématiques et Informatique, Faculté des Sciences Dhar Mahraz, BP 1796 Atlas, Fès, Morocco Received 13 June 2003; accepted 16 August 2004 Abstract We prove approximation and compactness results in inhomogeneous Orlicz-Sobolev spaces and lookat,asanapplication,theCauchy-Dirichletequation u ′ +A(u) +g(x,t,u, ∇u) =f ∈ W -1,x E M , where A is a Leray-Lions operator having a growth not necessarily of polynomial type.We also give a trace result allowing to deduce the continuity of the solutions with respect to time.  2004 Elsevier Ltd. All rights reserved. MSC: 35K15; 35K20; 35K60 Keywords: Inhomogeneous Orlicz–Sobolev spaces; Parabolic problems; Compactness 1. Introduction Let  be a bounded open subset of R N and let Q be the cylinder  × (0,T) with some given T> 0 and let A(u) =-div(a(x,t,u, ∇u)) be a Leray–Lions operator defined on L p (0,T ; W 1,p ()). ∗ Corresponding author. E-mail addresses: elmahi_abdelhak@yahoo.fr (A. Elmahi), meskinedriss@hotmail.com (D. Meskine). 0362-546X/$ - see front matter  2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2004.08.018