While working on an abstract problem, I came up with the following question:
Let $\Omega_1 := \mathrm B(-1, 1)$ and $\Omega_2 := \mathrm B(1, 1)$, where $\mathrm B(x, r) \subseteq \mathbb R^2$ denotes the ball with center $x \in \mathbb R^2$ and radius $r > 0$. Consider the heat equation $$u'(t) = \Delta u(t), \quad u(0) = u_0 \in L^p(\Omega_j), \quad j = 1, 2,$$ on both domains with Neumann boundary conditions, respectively, where $1 \leq p < \infty$. Then it is well known that both problems are well-posed, the solutions are given by strongly continuous semigroups and that the solutions converge uniformly to the equilibrium.
Now to the reason why I chose these specific domains. What happens if one considers mixed boundary conditions of a really specific type, namely, Neumann boundary conditions on $\partial \Omega_j \setminus \{0\}$ and a boundary condition in $0$ that models the behaviour that heat can pass through the "spatial singularity" in $0$ between the two domains? Would the solutions on both domains just average out over time or can this "singular nature" of the coupling of both phase spaces yield interesting effects besides the transfer that I would naively expect?
Moreover, is there any well-known literature on this specific question?
