Existence of a solution for a class of parabolic equations with three unbounded nonlinearities, natural growth terms and L1 data

We give an existence result of a renormalized solution for a class of nonlinear parabolic equations ∂b(x,u)∂t-div(a(x,t,u,∇u))+g(u)|∇u|p=f, where the right side belongs to L1(Ω × (0, T)), b(x, u) is an unbounded function of u and −div(a(x, t, u, ∇u)) is a Leray–Lions type operator with growth ∣∇u∣p−1 in ∇u, but without any growth assumption on u. The function g   is just assumed to be continuous on RR and satisfying a sign condition.