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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)

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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
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1 Answer

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
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