I am learning Mean value property(MVP) of heat equation . MVP of laplace equation was relatively easy to understand i think its because of the spherical symmetry . But i am not able to appreciate the MVP of heat equation. It's not very easy to imagine "heat ball". I would be glad for any kind of help. My question is how do i define a heat ball ? And how does it actually look like ? Text that i am following is as follows : http://www.math.ualberta.ca/~xinweiyu/527.1.08f/lec13.pdf ( theorem 9)
There is an illustration on page 53 of PDE by Evans. Nothing mysterious, just an ellipsoid-like shape with the "center" $(x,t)$ located at the center on the top boundary (not in the interior, as for elliptic PDE).
The definition is in the book you are reading, formula (23).