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.