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 $L^{p'}(0,T;W^{-1,p'}(Ω))$ and where b(x,u) is unbounded function of u and where $-div(a(x,t,u,∇u))$ is a Leray-Lions type operatorwith growth $|∇u|^{p-1}$ in ∇u. The critical growth condition on g is with respect to ∇u and no growth condition with respect to u, while the function $H(x,t,∇u)$ grows as $|∇u|^{p-1}$."