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.
