## Question on PDE

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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.

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When you say "the boundary for a such a heat equation is a circle" do you mean the level sets for a fundamental solution are circles? – S. Carnahan♦ Apr 25 2011 at 4:15
I think that is what I am implying. – unknown (google) Apr 25 2011 at 5:46

## 1 Answer

This answer makes some assumptions about what the OP is asking. In particular I am using Scott's interpretation that the boundaries of interest correspond to level sets of the fundamental solution.

I suppose you have to specify which level set you are talking about. Since the heat equation has infinite propagation speed, you can make the $t_i$ as close to zero as you like by looking at level sets $a_i = \epsilon$ for $\epsilon$ sufficiently small. If you fix $\epsilon$ (or $\epsilon_i$) then you are looking at a collection of three circles in the plane with radius $r_i(t) = \left(4kt \log(1/\epsilon) + 2kt \log(4 \pi k t)\right)^\frac{1}{2}$ (assuming the standard heat equation with diffusion constant k) and you are asking when these circles intersect. The circle about $A_i$ will intersect the circle about $A_j$ when $r_i + r_j \ge |A_i - A_j|$.

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Hi Aaron: Thank you for the answer. Can you write explicitly the system of PDEs that will start diffusing and stop at the closest neighbor's level set? – unknown (google) Apr 26 2011 at 1:54
I am wondering what the system would look like for more than $3$ points in $n \ge 3$ dimensions. – unknown (google) Apr 26 2011 at 1:55
Caveat: Maybe I'm not understanding your question.    I am imagining $u_t = D\Delta u$ where $u(x,t) \in \R^m$ and $x \in \R^n$ and $D$ is a diagonal matrix with entries $k_i$.  This is just a collection of uncoupled heat equations in $\R^n$ whose fundamental solution is just a vector of Gaussians.  It sounds like the solution that you are interested in corresponds to $u_i$ a Gaussian centered at $A_i$.  The stopping time is introduced artificially by picking levels $\epsilon_i$ and taking logarithms of the Gaussians to solve for the radius of the spherical level set as a function of time ... – Aaron Hoffman Apr 26 2011 at 2:24
(continued) ... and then call a collision any time that $|A_i - A_j| \ge r_i + r_j$.  Getting the PDE to stop diffusing without introducing an artificial stopping time would be more subtle.   – Aaron Hoffman Apr 26 2011 at 2:27
"Getting the PDE to stop diffusing without introducing an artificial stopping time would be more subtle." Actually I think I am interested in exactly this "automation" of the stopping times so that the description of the PDEs themselves would capture the closest neighbors. Such a description could provide an analytical gadget to some euclidean graph problems – unknown (google) Apr 26 2011 at 2:46
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