2.3. HEAT EQUATION 49

is in fact positive for all points x ∈

Rn

and times t 0. We interpret this

observation by saying the heat equation forces inﬁnite propagation speed

for disturbances. If the initial temperature is nonnegative and is positive

somewhere, the temperature at any later time (no matter how small) is

everywhere positive. (We will learn in §2.4.3 that the wave equation in

contrast supports ﬁnite propagation speed for disturbances.)

c. Nonhomogeneous problem. Now let us turn our attention to the

nonhomogeneous initial-value problem

(12)

ut − Δu = f in

Rn

× (0, ∞)

u = 0 on

Rn

× {t = 0}.

How can we produce a formula for the solution? If we recall the moti-

vation leading up to (9), we should note further that the mapping (x, t) →

Φ(x−y, t−s) is a solution of the heat equation (for given y ∈

Rn,

0 s t).

Now for ﬁxed s, the function

u = u(x, t; s) =

Rn

Φ(x − y, t − s)f(y, s) dy

solves

(12s)

ut(·; s) − Δu(·; s) = 0 in

Rn

× (s, ∞)

u(·; s) = f(·,s) on

Rn

× {t = s},

which is just an initial-value problem of the form (8), with the starting time

t = 0 replaced by t = s and g replaced by f(·,s). Thus u(·; s) is certainly

not a solution of (12).

However Duhamel’s

principle∗

asserts that we can build a solution of

(12) out of the solutions of (12s), by integrating with respect to s. The idea

is to consider

u(x, t) =

t

0

u(x, t; s) ds (x ∈

Rn,

t ≥ 0).

Rewriting, we have

(13)

u(x, t) =

t

0 Rn

Φ(x − y, t − s)f(y, s) dyds

=

t

0

1

(4π(t − s))n/2

Rn

e−

|x−y|

2

4(t−s)

f(y, s) dyds,

for x ∈

Rn,

t 0.

To conﬁrm that formula (13) works, let us for simplicity assume f ∈

C1

2(Rn

× [0, ∞)) and f has compact support.

∗Duhamel’s

principle has wide applicability to linear ODE and PDE and does not depend

on the speciﬁc structure of the heat equation. It yields, for example, the solution of the nonho-

mogeneous transport equation, obtained by diﬀerent means in §2.1.2. We will invoke Duhamel’s

principle for the wave equation in §2.4.2.