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}.
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