2.3. HEAT EQUATION 57 Infinite propagation speed again. The strong maximum principle im- plies that if U is connected and u ∈ C1 2(UT ) ∩ C( ¯ U T ) satisfies ⎧ ⎨ ⎩ ut − Δu = 0 in UT u = 0 on ∂U × [0,T ] u = g on U × {t = 0} where g ≥ 0, then u is positive everywhere within UT if g is positive some- where on U. This is another illustration of infinite propagation speed for disturbances. An important application of the maximum principle is the following uniqueness assertion. THEOREM 5 (Uniqueness on bounded domains). Let g ∈ C(ΓT ), f ∈ C(UT ). Then there exists at most one solution u ∈ C1 2(UT ) ∩ C( ¯T U ) of the initial/boundary-value problem (22) ut − Δu = f in UT u = g on ΓT . Proof. If u and ˜ u are two solutions of (22), apply Theorem 4 to w := ±(u − ˜). u We next extend our uniqueness assertion to the Cauchy problem, that is, the initial-value problem for U = Rn. As we are no longer on a bounded region, we must introduce some control on the behavior of solutions for large |x|. THEOREM 6 (Maximum principle for the Cauchy problem). Suppose u ∈ C1 2(Rn × (0,T ]) ∩ C(Rn × [0,T ]) solves (23) ut − Δu = 0 in Rn × (0,T ) u = g on Rn × {t = 0} and satisfies the growth estimate (24) u(x, t) ≤ Aea|x|2 (x ∈ Rn, 0 ≤ t ≤ T ) for constants A, a 0. Then sup Rn×[0,T ] u = sup Rn g.
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