Existence of a Solution for a Class of Parabolic Equations with Three Unbounded Nonlinearities

Abstract

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