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.
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