Existence and uniqueness of a solution for a class of parabolic equations with two unbounded nonlinearities

Abstract

In this paper we prove the existence and uniqueness of a renormalized solution for nonlinear parabolic equations whose model is \begin{eqnarray} \frac{\partial b(u)}{\partial t} - div\big(a(x,t,u,\nabla u)\big)=f+ div (g), \end{eqnarray} where the right side belongs to $L^{1}(Q)+L^{p'}(0,T;W^{-1,p'}(\Omega))$, where $b(u)$ is a real function of $u$ and where $-div(a(x,t,u,\nabla u))$ is a Leray-Lions type operator with growth $|\nabla u|^{p-1}$ in $\nabla u$, but without any growth assumption on $u$.