Existence of Renormalized Solutions for Nonlinear Parabolic Equations

Abstract

" We give an existence result of a renormalized solution for a class of nonlinear parabolic equations $$\\frac{\\partial b(x,u)}{\\partial t}-div(a(x,t,u,\\nabla u))+g(x,t,u,\\nabla u)+H(x,t,\\nabla u)=f,\\qquad in\\; Q_T,$$ where the right side belongs to $L^{p'}(0,T;W^{-1,p'}(\u03a9))$ and where b(x,u) is unbounded function of u and where $-div(a(x,t,u,\u2207u))$ is a Leray-Lions type operatorwith growth $|\u2207u|^{p-1}$ in \u2207u. The critical growth condition on g is with respect to \u2207u and no growth condition with respect to u, while the function $H(x,t,\u2207u)$ grows as $|\u2207u|^{p-1}$."