2.4. WAVE EQUATION 83

Cone of dependence

Proof. If ˜ u is another such solution, then w := u − ˜ u solves

⎧

⎨

⎩

wtt − Δw = 0 in UT

w = 0 on ΓT

wt = 0 on U × {t = 0}.

Deﬁne the “energy”

E(t) :=

1

2

U

wt

2(x,

t) + |Dw(x,

t)|2

dx (0 ≤ t ≤ T ).

We compute

˙

E (t) =

U

wtwtt + Dw · Dwt dx ˙=

d

dt

=

U

wt(wtt − Δw) dx = 0.

There is no boundary term since w = 0, and hence wt = 0, on ∂U × [0,T ].

Thus for all 0 ≤ t ≤ T , E(t) = E(0) = 0, and so wt,Dw ≡ 0 within UT .

Since w ≡ 0 on U × {t = 0}, we conclude w = u − ˜ u ≡ 0 in UT .

b. Domain of dependence. As another illustration of energy methods,

let us examine again the domain of dependence of solutions to the wave

equation in all of space. For this, suppose u ∈

C2

solves

utt − Δu = 0 in

Rn

× (0, ∞).

Fix x0 ∈

Rn,

t0 0 and consider the backwards wave cone with apex (x0,t0)

K(x0,t0) := {(x, t) | 0 ≤ t ≤ t0, |x − x0| ≤ t0 − t}.