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I am learning Mean value property (MVP) of the heat equation. MVP of Laplace equation was relatively easy to understand I think it is because of the spherical symmetry. But I am not able to appreciate the MVP of heat equation. It's not very easy to imagine the "heat ball".

I would be glad for any kind of help.

My questions are

  • How do I define a heat ball?
  • How does it actually look like?

Text that i am following is as follows : (theorem 9)

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What is your question? – user31373 Jun 22 '12 at 15:09
@LeonidKovalev : My question is how do i define a heat ball ? And how does it actually look like ? – Theorem Jun 22 '12 at 15:26
Could you give a reference to a text you are reading? – abatkai Jun 22 '12 at 15:31
@abatkai : I have added the reference . Thank you – Theorem Jun 22 '12 at 15:40

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).

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Sir, Why should it be ellipsoid ? Can you help me to understand it – Theorem Jun 22 '12 at 18:07
@Theorem For each fixed value of time variable $s$, you get a two-dimensional slice which is a circle. The radius of the circle depends on $s$: it drops to zero when $s=t$ and when $s$ is much smaller than $t$. – user31373 Jun 22 '12 at 18:35
Also, -1 for lack of own effort. – user31373 Jun 22 '12 at 18:36

The "heat ball" is defined as it is in the note you cited: $$ E(x,t;r)=\{(y,s)\in {\Bbb R}^{n+1}\mid s\leq t,\ \dfrac{1}{(4\pi(t-s))^{n/2}}e^{-\frac{|x-y|^2}{4(t-s)}}\geq\frac{1}{r^n}\}. $$ The Wikipedia article Mean-value property for the heat equation also gives a similar definition.

To get some ideas of what such "ball" would look like, consider $n=1$ and $$ E(0,0;1)=\{(y,s)\in{\Bbb R}^2\mid 0\leq s\leq\frac{1}{4\pi}, |y|^2\leq 2s\log\frac{1}{4\pi s}\} $$ Using the $s$-axis as the horizontal one, you can see the picture of $E(0,0;1)$:

enter image description here

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