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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.

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