Existence of a renormalized solutions for parabolic–elliptic system in anisotropic Orlicz–Sobolev spaces

The purpose of this paper is to establish the existence of renormalized solutions to the following parabolic–elliptic system \(\begin{aligned} \varvec{\left\{ \begin{aligned} \frac{\partial u}{\partial t}-\sum _{i=1}^d \partial _i\left( a_i\left( x, t, u, \partial _i u\right) \right)&=\kappa (u)|\nabla v|^{2}{} & {} \text{ in } Q_{T}=\Omega \times (0, T), \\ {\text {div}}(\kappa (u) \nabla v)+{\text {div}} F(u)&=0{} & {} \text{ in } Q_{T}, \\ u&=0{} & {} \text{ on } \partial \Omega \times (0, T), \\ v&=0{} & {} \text{ on } \partial \Omega \times (0, T), \\ u(\cdot , 0)&=u_{0}{} & {} \text{ in } \Omega , \end{aligned}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{ccc}\hfill \frac{\mathit{\partial }\mathit{u}}{\mathit{\partial }\mathit{t}}\mathbf{-}\mathbf{\sum }_{\mathit{i}\mathbf{=}\mathbf{1}}^{\mathit{d}}{\mathit{\partial }}_{\mathit{i}}\left({\mathit{a}}_{\mathit{i}},\left(\mathit{x},\mathbf{,},\mathit{t},\mathbf{,},\mathit{u},\mathbf{,},{\mathit{\partial }}_{\mathit{i}},\mathit{u}\right)\right)& \mathbf{=}{\mathit{\kappa }\mathbf{(}\mathit{u}\mathbf{)}\mathbf{|}\mathbf{\nabla }\mathit{v}\mathbf{|}}^{\mathbf{2}}\hfill & \hfill \text{in}{\mathit{Q}}_{\mathit{T}}\mathbf{=}\mathbf{\Omega }\mathbf{\times }\mathbf{(}\mathbf{0}\mathbf{,}\mathit{T}\mathbf{)}\mathbf{,}\\ \hfill \text{div}\mathbf{(}\mathit{\kappa }\mathbf{(}\mathit{u}\mathbf{)}\mathbf{\nabla }\mathit{v}\mathbf{)}\mathbf{+}\text{div}\mathit{F}\mathbf{(}\mathit{u}\mathbf{)}& \mathbf{=}\mathbf{0}\hfill & \hfill \text{in}{\mathit{Q}}_{\mathit{T}}\mathbf{,}\\ \hfill \mathit{u}& \mathbf{=}\mathbf{0}\hfill & \hfill \text{on}\mathit{\partial }\mathbf{\Omega }\mathbf{\times }\mathbf{(}\mathbf{0}\mathbf{,}\mathit{T}\mathbf{)}\mathbf{,}\\ \hfill \mathit{v}& \mathbf{=}\mathbf{0}\hfill & \hfill \text{on}\mathit{\partial }\mathbf{\Omega }\mathbf{\times }\mathbf{(}\mathbf{0}\mathbf{,}\mathit{T}\mathbf{)}\mathbf{,}\\ \hfill \mathit{u}\mathbf{(}\mathbf{·}\mathbf{,}\mathbf{0}\mathbf{)}& \mathbf{=}{\mathit{u}}_{\mathbf{0}}\hfill & \hfill \text{in}\mathbf{\Omega }\mathbf{,}\end{array}\right)\end{array}$$ We will consider the inhomogeneous anisotropic Orlicz–Sobolev spaces, without imposing the \(\Delta _{2}\) ${\mathrm{\Delta }}_2$ -condition on the N-functions.