On a second order linear parabolic equation with variable coefficients in a non-regular domain of $\mathbb{R}^{3}$

This paper is devoted to the study of the following variable-coefficient parabolic equation in non-divergence form
$$\begin{equation*} \partial _{t}u-\sum_{i=1}^{2}a_{i}(t,x_{1},x_{2})\partial_{ii}u+\sum_{i=1}^{2}b_{i}(t,x_{1},x_{2})\partial _{i}u+c(t,x_{1},x_{2})u=f(t,x_{1},x_{2}), \end{equation*}$$
subject to Cauchy–Dirichlet boundary conditions. The problem is set in a non-regular domain of the form
$$\begin{equation*} Q=\left\{ \left( t,x_{1}\right) \in\mathbb{R}^{2}:0<t<T, \varphi _{1}\left( t\right) <x_{1}<\varphi _{2}\left( t\right)\right\} \times \left] 0,b\right[, \end{equation*}$$
where $\varphi _{k},\; k=1,2$ are "smooth" functions. One of the main issues of this work is that the domain can possibly be non-regular, for instance, the singular case where $\varphi _{1}$ coincides with $\varphi_{2}$ for $t=0$ is allowed. The analysis is performed in the framework of anisotropic Sobolev spaces by using the domain decomposition method. This work is an extension of the constant-coefficients case studied in [15].