Renormalized Solution for the Nonlinear Parabolic Problem with Lower Order Terms

Abstract

In this paper, we consider the following problem: \[ \begin{cases} -\nabla\cdot A(x,u,\nabla u) + H(x,u,\nabla u) = f(x), & x \in \Omega, u = 0, & x \in \partial \Omega, \end{cases} \] in a bounded open set \( \Omega \subset \mathbb{R}^N \). We have established certain gradient estimates and proved the existence of a renormalized solution for the equation.