Homogenization of a parabolic equation in perforated domain with Dirichlet boundary condition (Q1857382)

This paper is devoted to the homogenization of the family of parabolic equations over periodically perforated domains \[ \begin{cases} \partial_tb\left(\frac{x}{d_\varepsilon},u_\varepsilon\right)-\text{div }a(u_\varepsilon,\nabla u_\varepsilon)=f(x,t) & \text{in }\Omega_\varepsilon\times (0,T),\\ u_\varepsilon=0 & \text{on }\partial\Omega_\varepsilon\times(0,T),\\ u_\varepsilon(x,0)=u_0(x) & \text{in }\Omega_\varepsilon,\end{cases} \tag{\(1_\varepsilon\)} \] where \(\Omega_\varepsilon=\Omega\setminus S_\varepsilon\) is a periodically perforated domain and \(d_\varepsilon\) is a sequence of positive numbers which goes to zero. The authors' goal is to study the limiting behaviour of solution \((1_\varepsilon)\) and to obtain the limiting equation satisfied by the limit.