This article will focus on establishing the existence of renormalized solution for the subsequent parabolic equation \(\begin{aligned} \left\{ \begin{aligned} \frac{\partial w}{\partial t}+\textrm{H}(w)&=\frac{\beta \kappa (w)}{\left( \displaystyle \int _{\Omega }\kappa (w)\textrm{dx}\right) ^{2}} & \text{ in } Q=\Omega \times (0, T), \\ w&=0 & \text{ on } \Gamma =\partial \Omega \times (0, T), \\ w(\cdot , 0)&=w_{0} & \text{ in } \Omega , \end{aligned}\right. \end{aligned}\) In this context, the operator \(\textrm{H}(w)=-\operatorname {div}{\mathcal {H}}(x,\tau ,w,\nabla w)\) corresponds to a Leray-Lions operator defined on the inhomogeneous Orlicz–Sobolev space \(W_0^{1,x} L_A(Q )\) into its dual \(W^{-1,x} L_{{\bar{A}}}(Q )\) , A and \({\bar{A}}\) are two N-functions related to the growth of \({\mathcal {H}}\) .
