Existence of a renormalized solutions to a nonlinear system in Orlicz spaces

Abstract

In this paper, we will be concerned with the existence of renormalized solutions to the following parabolic-elliptic system$$\left\{\begin{aligned}\frac{\partial u}{\partial t}+Au&=\sigma(u)|\nabla \varphi|^{2} & & \text { in } Q_{T}=\Omega \times(0, T), \\-\text{div}(\sigma(u) \nabla \varphi) &=\text{div} F(u) & & \text { in } Q_{T}, \\u &=0 & & \text { on } \partial \Omega \times(0, T), \\\varphi &=0 & & \text { on } \partial \Omega \times(0, T), \\u(\cdot, 0) &=u_{0} & & \text { in } \Omega,\end{aligned}\right.$$where $Au=-\text{div }a(x ,t , u, \nabla u)$ is a Leray-Lions operator defined on the inhomogeneous Orlicz-Sobolev space $W_0^{1,x} L_M(Q_T )$ into its dual, $M$ is a N-function related to the growth of $a$. $M$ does not satisfy the $\Delta_{2}$-condition, and $\sigma$ and $F$ are two Carath\'{e}odory functions defined in $Q_{T}\times\mathbb{R}$.\\