2.3. HEAT EQUATION 53
A “heat ball”
This is a region in space-time, the boundary of which is a level set of
Φ(x−y, t−s). Note that the point (x, t) is at the center of the top. E(x, t; r)
is sometimes called a “heat ball”.
THEOREM 3 (A mean-value property for the heat equation). Let u
C1
2(UT
) solve the heat equation. Then
(19) u(x, t) =
1
4rn
E(x,t; r)
u(y, s)
|x
y|2
(t s)2
dyds
for each E(x, t; r) UT .
Formula (19) is a sort of analogue for the heat equation of the mean-value
formulas for Laplace’s equation. Observe that the right-hand side involves
only u(y, s) for times s t. This is reasonable, as the value u(x, t) should
not depend upon future times.
Proof. Shift the space and time coordinates so that x = 0 and t = 0. Upon
mollifying if necessary, we may assume u is smooth. Write E(r) = E(0, 0; r)
and set
(20)
φ(r) :=
1
rn
E(r)
u(y, s)
|y|2
s2
dyds
=
E(1)
u(ry,
r2s)
|y|2
s2
dyds.
We compute
φ (r) =
E(1)
n
i=1
uyi yi
|y|2
s2
+ 2rus
|y|2
s
dyds
=
1
rn+1
E(r)
n
i=1
uyi yi
|y|2
s2
+ 2us
|y|2
s
dyds
=: A + B.
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