Renormalized Solution for the Nonlinear Parabolic Problem with Lower Order Terms

TL;DR

This paper establishes the existence and uniqueness of renormalized solutions for a nonlinear parabolic PDE with lower order terms in bounded domains, using advanced mathematical techniques.

Contribution

It introduces a novel approach to prove well-posedness of nonlinear parabolic equations with non-coercive terms in bounded domains.

Findings

Proved existence of renormalized solutions for the PDE.

Established uniqueness of these solutions.

Developed gradient estimates for solutions.

Abstract

In this paper, we consider the following nonlinear parabolic equation with non-coercive terms in \(R^N\) space \[ \dfrac{\partial u}{\partial t} -\nabla \cdot (a(x,t,u,\nabla u)+ \Phi(x,t,\nabla u))=f, \text{ in }\Omega \times (0,T). \] Here \(\Omega\) is a bounded open set of \(R^N\) with the boundary \(\partial \Omega\) satisfying Lipschitz condition. The Carath\'eodory function \(\Phi\) is restricted by $|\Phi(x,t,s)|\le c(x,t)|s|^\gamma$ with parameters depending on $p$ and $N$. And the initial value $u(x,0)=u_0(x)$. For convenience, we define the domain $Q := \Omega \times (0,T)$ and the boundary similarly. Then for $f\in L^1(Q)$ and $u_0\in L^1(\Omega)$, we prove the existence and uniqueness of a renormalized solution via truncation methods, monotone operator theory, and a prior gradient estimates.