80 2. FOUR IMPORTANT LINEAR PDE

THEOREM 3 (Solution of wave equation in even dimensions). Assume n

is an even integer, n ≥ 2, and suppose also g ∈

Cm+1(Rn),

h ∈

Cm(Rn),

for m =

n+2

2

. Deﬁne u by (38). Then

(i) u ∈

C2(Rn

× [0, ∞)),

(ii) utt − Δu = 0 in

Rn

× (0, ∞),

and

(iii) lim

(x,t)→(x0,0)

x∈Rn, t0

u(x, t) =

g(x0),

lim

(x,t)→(x0,0)

x∈Rn, t0

ut(x, t) =

h(x0)

for each point

x0

∈

Rn.

This follows from Theorem 2. Observe, in contrast to formula (31), that

to compute u(x, t) for even n we need information on u = g, ut = h on all

of B(x, t) and not just on ∂B(x, t).

Huygens’ principle. Comparing (31) and (38), we observe that if n is odd

and n ≥ 3, the data g and h at a given point x ∈

Rn

aﬀect the solution u only

on the boundary {(y, t) | t 0, |x − y| = t} of the cone C = {(y, t) | t 0,

|x−y| t}. On the other hand, if n is even, the data g and h aﬀect u within

all of C. In other words, a “disturbance” originating at x propagates along

a sharp wavefront in odd dimensions, but in even dimensions it continues

to have eﬀects even after the leading edge of the wavefront passes. This is

Huygens’ principle.

2.4.2. Nonhomogeneous problem.

We next investigate the initial-value problem for the nonhomogeneous

wave equation

(39)

utt − Δu = f in

Rn

× (0, ∞)

u = 0, ut = 0 on

Rn

× {t = 0}.

Motivated by Duhamel’s principle (introduced earlier in §2.3.1), we deﬁne

u = u(x, t; s) to be the solution of

(40s)

utt(·; s) − Δu(·; s) = 0 in

Rn

× (s, ∞)

u(·; s) = 0, ut(·; s) = f(·,s) on

Rn

× {t = s}.

Now set

(41) u(x, t) :=

t

0

u(x, t; s) ds (x ∈

Rn,t

≥ 0).

Duhamel’s principle asserts this is a solution of

(42)

utt − Δu = f in

Rn

× (0, ∞)

u = 0, ut = 0 on

Rn

× {t = 0}.