Renormalized Solution for the Nonlinear Parabolic Problem with Lower Order Terms

Abstract

In this paper, we consider the following nonlinear parabolic equation with non-coercive terms in \(R^N\) space \[ \dfrac{\partial u}{\partial t} -\nabla \cdot (a(x,t,u,\nabla u)+ Φ(x,t,\nabla u))=f, \text{ in }Ω\times (0,T). \] Here \(Ω\) is a bounded open set of \(R^N\) with the boundary \(\partial Ω\) satisfying Lipschitz condition. The Carathéodory function \(Φ\) is restricted by $|\hat{I}¦(x,t,s)|\le c(x,t)|s{|}^3$ with parameters depending on $p$ and $N$. And the initial value $u(x,0)=u_0(x)$. For convenience, we define the domain $Q:=\hat{I}\text{c◯}\times (0,T)$ and the boundary similarly. Then for $f\in L^1(Q)$ and $u_0\in L^1(\hat{I}\text{c◯})$, we prove the existence and uniqueness of a renormalized solution via truncation methods, monotone operator theory, and a prior gradient estimates.