2.3. HEAT EQUATION 63 THEOREM 10 (Uniqueness). There exists only one solution u ∈ C1 2( ¯T U ) of the initial/boundary-value problem (40). Proof. 1. If ˜ u is another solution, w := u − ˜ u solves (41) wt − Δw = 0 in UT w = 0 on ΓT . 2. Set e(t) := U w2(x, t) dx (0 ≤ t ≤ T ). Then ˙(t) e = 2 U wwt dx ˙= d dt = 2 U wΔw dx = −2 U |Dw|2 dx ≤ 0, and so e(t) ≤ e(0) = 0 (0 ≤ t ≤ T ). Consequently w = u − ˜ u ≡ 0 in UT . Observe that the foregoing is a time-dependent variant of the proof of Theorem 16 in §2.2.5. b. Backwards uniqueness. A rather more subtle question asks about uniqueness backwards in time for the heat equation. For this, suppose u and ˜ u are both smooth solutions of the heat equation in UT , with the same boundary conditions on ∂U: (42) ut − Δu = 0 in UT u = g on ∂U × [0,T ], (43) ˜t u − Δ˜ u = 0 in UT ˜ u = g on ∂U × [0,T ], for some function g. Note carefully that we are not supposing u = ˜ u at time t = 0.
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