Nonlinear parabolic equations with p-growth and unbounded data

The purpose of this paper is to prove the existence of a solution for a non linear parabolic equation in the form u(t) - div(a(t, x, u, Du)) = H(t, x, u, Du) - div(g(t, x)) in Q(T) =]0,T[x Omega, Omega subset of R-N, With an i nitial condition u(0) = u(0), where u(0) is not bounded, \H(t, x, u, xi)\ l ess than or equal to beta\xi\(p) + f(t, x) + beta e(lambda 1\u\), f, \g\(p/ (p-1)) is an element of L-r(Q(T)) for some r = r(N) greater than or equal t o 1, and - div(a(t, x, u, Du)) is the usual Leray-Lions operator. (C) Acade mie des Sciences/Elsevier, Paris.