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 $$\dfrac{\partial u}{\partial t} -\nabla \cdot (a(x,t,u,\nabla u)+ \Phi(x,t,\nabla u))=f, \text{ in }\Omega \times (0,T).$$ 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.