Renormalized Solution for the Nonlinear Parabolic Problem with Lower Order Terms

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