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}.
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