### Heat ball

There are very few things I find unsatisfactory in L.C. Evans’ wonderful textbook on Partial Differential Equations; one of them is the illustration (on p.53 of the second edition) of the “heat ball”.

The heat ball is a region with respect to which an analogue of the mean value property of solutions to Laplace’s equation can be expressed, now for solutions of the heat equation. In the case of the Laplace’s equation, the regions are round balls. In the case of the heat equation, the regions are somewhat more complicated. They are defined by the expression

$\displaystyle E(x,t;r) := \left\{ (y,s)\in \mathbb{R}^{n+1}~|~s \leq t, \Phi(x-y, t-s) \geq \frac{1}{r^n} \right\}$

where $\Phi$ is the fundamental solution of the heat equation

$\displaystyle \Phi(x,t) := \frac{1}{(4\pi t)^{n/2}} e^{- \frac{|x|^2}{4t}}.$

In the expressions above, the constant $n$ is the number of spatial dimensions; $r$ is the analogue of the radius of the ball, and in $E(x,t;r)$, the point $(x,r)$ is the center. Below is a better visualization of the heat balls: the curves shown are the boundaries $\partial E(0,5;r)$ in dimension $n = 1$, for radii between 0.75 and 4 in steps of 0.25 (in particular all the red curves have integer radii). In higher dimensions the shape is generally the same, though they appear more “squashed” in the $t$ direction.

1-dimensional heat balls centered at (0,5) for various radii. (Made using Desmos)