Renormalized solutions for nonlinear anisotropic parabolic equations

Abstract

In this paper, we establish the existence of a renormalized solutions of anisotropic parabolic operators of the type $\frac{\partial b(x,u)}{\partial t}-\sum\limits_{i=1}^{N}\partial_{i}a_{i}(x, t, u, \nabla u)=f$ and $b(x, u)(t\ =\ 0) = b(x, u_{0})$, where the right hand side $f$’ belongs to $L^{1}(Q_{T})$ and $b(x, u_{0})$ belongs to $L^{1}(\Omega)$. The function $b(x, u)$ is unbounded on u, the operator $-\sum_{i=1}^{N}\partial_{i}a_{i}(x, t, u, \nabla u)$ is a Leray-Lions operator.