I am wondering if there exists some application of the following classical result (I write the version that appears in Evans' book):
Theorem: Let $u \in C_{1}^{2}(U_{T})$ solve the heat equation. Then $$u(x,t)=\frac{1}{4r^{n}} \iint_{E(x,t;r)}u(y,s) \frac{|x-y|^{2}}{(t-s)^{2}}dyds$$ for each $E(x,t;r) \subset U_{T}$.
With application I mean something different from the theorems that Evans proves using this formula, and also something interesting but not to large or difficult, because my intention is to present it to my students in a PDE class.
What I did was addapt some "Harnack type theorems": if a sequence of functions satisfy the heat equation, so does its limit (in a good enough sense). This is an easy consequence of the mean-value property: you just pass the limit to the inside of the integral.
This idea come to me when I was reading Watson's Introduction to Heat Potential Theory. Indeed, what I mentioned in the previous paragraph is a particular case of Theorem 1.31 of this book.