46 2. FOUR IMPORTANT LINEAR PDE

for r = |y|, =

d

dr

. Now if we set α =

n

2

, this simpliﬁes to read

(rn−1w

) +

1

2

(rnw)

= 0.

Thus

rn−1w

+

1

2

rnw

= a

for some constant a. Assuming w, w → 0 fast enough as r → ∞, we

conclude a = 0; whence

w = −

1

2

rw.

But then for some constant b

(7) w =

be−

r

2

4

.

Combining (4), (7) and our choices for α, β, we conclude that

b

tn/2

e−

|x|

2

4t

solves the heat equation (1).

This computation motivates the following

DEFINITION. The function

Φ(x, t) :=

1

(4πt)n/2

e−

|x|2

4t

(x ∈

Rn,

t 0)

0 (x ∈

Rn,

t 0)

is called the fundamental solution of the heat equation.

Notice that Φ is singular at the point (0, 0). We will sometimes write

Φ(x, t) = Φ(|x|,t) to emphasize that the fundamental solution is radial in

the variable x. The choice of the normalizing constant

(4π)−n/2

is dictated

by the following

LEMMA (Integral of fundamental solution). For each time t 0,

Rn

Φ(x, t) dx = 1.

Proof. We calculate

Rn

Φ(x, t) dx =

1

(4πt)n/2

Rn

e−

|x|

2

4t

dx

=

1

πn/2

Rn

e−|z|2

dz

=

1

πn/2

n

i=1

∞

−∞

e−zi

2

dzi = 1.