84 2. FOUR IMPORTANT LINEAR PDE

THEOREM 6 (Finite propagation speed). If u ≡ ut ≡ 0 on B(x0,t0) ×

{t = 0}, then u ≡ 0 within the cone K(x0,t0).

In particular, we see that any “disturbance” originating outside B(x0,t0)

has no eﬀect on the solution within K(x0,t0) and consequently has ﬁnite

propagation speed. We already know this from the representation formulas

(31) and (38), at least assuming g = u and h = ut on

Rn

× {t = 0} are suf-

ﬁciently smooth. The point is that energy methods provide a much simpler

proof.

Proof. Deﬁne the local energy

e(t) :=

1

2

B(x0,t0−t)

ut

2(x,

t) + |Du(x,

t)|2

dx (0 ≤ t ≤ t0).

Then

(46)

˙(t) e =

B(x0,t0−t)

ututt + Du · Dut dx −

1

2

∂B(x0,t0−t)

ut

2

+

|Du|2

dS

=

B(x0,t0−t)

ut(utt − Δu) dx

+

∂B(x0,t0−t)

∂u

∂ν

ut dS −

1

2

∂B(x0,t0−t)

ut

2

+

|Du|2

dS

=

∂B(x0,t0−t)

∂u

∂ν

ut −

1

2

ut

2

−

1

2

|Du|2

dS.

Now

(47)

∂u

∂ν

ut ≤ |ut||Du| ≤

1

2

ut

2

+

1

2

|Du|2,

by the Cauchy–Schwarz and Cauchy inequalities (§B.2). Inserting (47) into

(46), we ﬁnd ˙(t) e ≤ 0; and so e(t) ≤ e(0) = 0 for all 0 ≤ t ≤ t0. Thus ut,

Du ≡ 0, and consequently u ≡ 0 within the cone K(x0,t0).

A generalization of this proof to more complicated geometry appears

later, in §7.2.4. See also §12.1 for a similar calculation for a nonlinear wave

equation.

2.5. PROBLEMS

In the following exercises, all given functions are assumed smooth, unless

otherwise stated.