Let $G_0$ be a set in $\mathbb{R}^n$, $n=2,3$, for any factor $\alpha >0$ denote $\alpha G_0=\{x \in \mathbb{R}^n, x/\alpha \in G_0\}$, for any $i \in \mathbb{Z}^n$ denote $G_0+i=\{x \in \mathbb{R}^n: x-i \in G_0\}$, $G_0+\mathbb{Z}^n = \cup_{i \in \mathbb{Z}^n} (G_0+i)$. Introduce two small parameters $\epsilon, \delta >0$ and a great parameter $\alpha >0$.
Let $\Omega_{\delta}= \delta G_0 +\mathbb{Z}^n$, where $G_0$ is a bounded domain with Lipschitzian boundary, $\Omega_{\epsilon,\delta}= \epsilon \Omega_{\delta}$.
Define a function $q_{\delta}: \mathbb{R}^n \to \mathbb{R}$ by $$ q_{\delta}(\xi) = \begin{cases} 1, & \xi \in \Omega_{\delta};\\ 0, & x \notin \Omega_{\delta}. \end{cases} $$ We consider the parabolic equation \begin{equation}\label{1} \dfrac{\partial u_{\epsilon,\delta}}{\partial t} - \Delta u_{\epsilon, \delta} + \omega q_{\delta}(\dfrac{x}{\epsilon}) u_{\epsilon,\delta} = f(x,t) \end{equation} in $\mathbb{R}^n$ with the initial condition \begin{equation}\label{2} u_{\epsilon,\delta}(x,0)=0, \end{equation} where $f(x,t) \in C^2(\mathbb{R}^n \times \overline{R}_+)$ is an $1$-periodic function in $x$.
My question is how we can prove the existence of solution of the parabolic equetion wuth the initial condition? I try to use Schauder, but wit no result. Help me please