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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" in the following theorem from a note:

enter image description here

Here are questions:

  • How do I define a heat ball?
  • How does it actually look like?
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    What is your question?2012-06-22
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    @LeonidKovalev : My question is how do i define a heat ball ? And how does it actually look like ?2012-06-22
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    Could you give a reference to a text you are reading?2012-06-22
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    @abatkai : I have added the reference . Thank you2012-06-22

2 Answers 2

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The "heat ball" is defined as it is in the note you cited which is bases on Evans's Partial Differential Equations Chapter 2.3.

For fixed $x\in{\bf R}^n$, $t\in{\bf R}$ and $r>0$, we define $$ E(x,t;r)=\left\{(y,s)\in {\bf R}^{n+1}\bigg|\; s\leqslant t,\ \dfrac{1}{(4\pi(t-s))^{n/2}}\exp\left({-\dfrac{|x-y|^2}{4(t-s)}}\right)\geqslant\frac{1}{r^n}\right\}. $$

The Wikipedia article Mean-value property for the heat equation also gives a similar definition.


Note that in the definition, one should replace $s\leqslant t$ with $s

The boundary of the heat ball is like this:

enter image description here

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    How did you derive the expression for $E(0,0; 1)$? If you assume $t = 0$, shouldn't $s$ be less than $0$?2016-09-14
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    @el_tenedor: Thanks for pointing that out! That's certainly a mistake and I have now edited the answer.2017-08-11
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    I have just noticed that I'm responding to a comment about one year old.2017-08-11
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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 it2012-06-22
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    @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 $t$.2012-06-22