Renormalized Solution for the Nonlinear Parabolic Problem with Lower Order Terms

TL;DR

This paper establishes the existence of renormalized solutions for a nonlinear parabolic PDE with lower order terms, providing gradient estimates in a bounded domain.

Contribution

It introduces a method to prove existence of solutions for complex nonlinear PDEs with lower order terms, expanding the theoretical understanding.

Findings

Established gradient estimates for the PDE

Proved existence of renormalized solutions

Extended solution concepts to nonlinear problems 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.