64 2. FOUR IMPORTANT LINEAR PDE THEOREM 11 (Backwards uniqueness). Suppose u, ˜ u C2( ¯T U ) solve (42), (43). If u(x, T ) = ˜(x, u T ) (x U), then u ˜ u within UT . In other words, if two temperature distributions on U agree at some time T 0 and have had the same boundary values for times 0 t T , then these temperatures must have been identically equal within U at all earlier times. This is not at all obvious. Proof. 1. Write w := u ˜ u and, as in the proof of Theorem 10, set e(t) := U w2(x, t) dx (0 t T ). As before (44) ˙(t) e = −2 U |Dw|2 dx ˙= d dt . Furthermore (45) ¨(t) e = −4 U Dw · Dwt dx = 4 U Δwwt dx = 4 U (Δw)2 dx by (41). Now since w = 0 on ∂U, U |Dw|2 dx = U wΔw dx U w2 dx 1/2 U (Δw)2 dx 1/2 . Thus (44) and (45) imply (˙(t))2 e = 4 U |Dw|2 dx 2 U w2 dx 4 U (Δw)2 dx = e(t)¨(t). e
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