Existence Results for a Class of Nonlinear Parabolic Equations in Orlicz Spaces

is a LerayLions operator defined on W 1,x 0 LM (Ω), M is an appropriate N -function and which grows like M̄M(β K |∇u|) with respect to ∇u, but which is not restricted by any growth condition with respect to u (see assumptions (3.3)-(3.6)). The function Φ is just assumed to be continuous on R. Under these assumptions, the above problem does not admit, in general, a weak solution since the fields a(x, t, u,∇u) and Φ(u) do not belong in (Lloc(Q) N in general. To overcome this difficulty we use in this paper the framework of renormalized solutions. This notion was introduced by Lions and DiPerna [31] for the study of Boltzmann equation (see also [27], [11], [29], [28], [2]). A large number of papers was devoted to the study the existence of renormalized solution of parabolic problems under various assumptions and in different contexts: for a review on classical results see [7], [30], [9], [8], [4], [5], [34], [12], [13], [14]. The existence and uniqueness of renormalized solution of (1.1)-(1.3) has been proved in H. Redwane [34, 35] in the case where Au = −div (