46 2. FOUR IMPORTANT LINEAR PDE
for r = |y|, =
. Now if we set α =
, this simpliﬁes to read
for some constant a. Assuming w, w → 0 fast enough as r → ∞, we
conclude a = 0; whence
w = −
But then for some constant b
(7) w =
Combining (4), (7) and our choices for α, β, we conclude that
solves the heat equation (1).
This computation motivates the following
DEFINITION. The function
Φ(x, t) :=
0 (x ∈
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
by the following
LEMMA (Integral of fundamental solution). For each time t 0,
Φ(x, t) dx = 1.
Proof. We calculate
Φ(x, t) dx =
dzi = 1.