I am looking for a reference where the following situation or something similar could have been studied. As a foreword, my question may not be very technical since I am from an engineering background. I have tried to provide an intuitive explanation of the problem. I am also looking for a mildly technical reference or answer:

Heat equation can be used to study diffusion of heat on a surface. On a plane the boundary for a such a heat equation is a circle. I am looking for a system of three heat equation type PDEs (call them $a_{1}$, $a_{2}$ and $a_{3}$) so that some conditions are satisfied.

1) $a_{1}$, $a_{2}$ and $a_{3}$ describe propagation of heat starting at three different points $A_{1}$, $A_{2}$ and $A_{3}$ on the plane.

2) Stopping time of $a_{i}$ is when intersection of boundary of $a_{i}$ and union of boundaries $a_{j}$ and $a_{k}$ is non empty with $i \ne j \ne k \ne i$ $\forall i \in \{1,2,3\}$.

Let the stopping time of $a_{i}$ be $t_{i}$. After giving a simple description of pdes that could satisfy the above conditions, I also need to find expressions for $t_{i}$ which is what I am truly after since it finding $t_{i}$ could provide distance between $A_{i}$ to it closest neighbor without using the euclidean formula. Can one get the PDE to stop diffusing without introducing an artificial stopping time? Can one generalize this to many points in $n$-dimensions (real or complex). Such a system could capture closest neighbors to each given point.