# Splitting a PDE into two subdomains: Regularity conditions at the interface point(s)

My aim is to split a given problem into two nonoverlapping subdomains.

Let us, for example, consider the initial-boundary value problem: Given $$u_0 \in L_2(0,1)$$ and $$f \in L_2(0,1)$$, find $$u$$ such that

$$\begin{eqnarray} \begin{cases} u_t - u_{xx} &=& f(x), \quad (x,t) \in (0,1) \times \{t > 0\}, \\ u(x,0) &=& u_0(x), \quad x \in (0,1) \end{cases} \end{eqnarray}$$

with non-homogeneous Dirichlet boundary conditions, let's say $$u(0,t) = a$$, $$u(1,t) = b$$, $$a,b \in \mathbb{R}$$.

Next, consider the following split problem:

$$\begin{eqnarray} \begin{cases} \mathcal{L}_t u_1 &=& f(x), \quad (x,t) \in (0,\frac{1}{2}) \times \{t > 0\}, \\ u_1(x,0) &=& u_0(x), \quad x \in (0,\frac{1}{2}) \\ u_1(0,t) &=& a, \quad t > 0 \\ u_1(\frac{1}{2},t) &=& u_2(\frac{1}{2},t), \quad t > 0 \end{cases} \end{eqnarray}$$

and

$$\begin{eqnarray} \begin{cases} \mathcal{L}_t u_2 &=& f(x), \quad (x,t) \in (\frac{1}{2},1) \times \{t > 0\}, \\ u_2(x,0) &=& u_0(x), \quad x \in (\frac{1}{2},1) \\ u_2(1,t) &=& b, \quad t > 0 \\ u_2(\frac{1}{2},t) &=& u_1(\frac{1}{2},t), \quad t > 0 \end{cases} \end{eqnarray}$$ where $$\mathcal{L}_t := \frac{\partial}{\partial t} - \frac{\partial^2}{\partial x^2}$$.

My question is what kind of regularity condition(s) for $$u_1$$ and $$u_2$$, should I impose at the point $$x = \frac{1}{2}$$. Also resources related to this subject are welcome.

Thank you for your time.