2.3. HEAT EQUATION 47 A different derivation of the fundamental solution of the heat equation appears in §4.3.1. b. Initial-value problem. We now employ Φ to fashion a solution to the initial-value (or Cauchy) problem (8) ut Δu = 0 in Rn × (0, ∞) u = g on Rn × {t = 0}. Let us note that the function (x, t) Φ(x, t) solves the heat equation away from the singularity at (0, 0), and thus so does (x, t) Φ(x y, t) for each fixed y Rn. Consequently the convolution (9) u(x, t) = Rn Φ(x y, t)g(y) dy = 1 (4πt)n/2 Rn e− |x−y| 2 4t g(y) dy (x Rn, t 0) should also be a solution. THEOREM 1 (Solution of initial-value problem). Assume g C(Rn) L∞(Rn), and define u by (9). Then (i) u C∞(Rn × (0, ∞)), (ii) ut(x, t) Δu(x, t) = 0 (x Rn, t 0), and (iii) lim (x,t)→(x0,0) x∈Rn, t0 u(x, t) = g(x0) for each point x0 Rn. Proof. 1. Since the function 1 tn/2 e− |x| 2 4t is infinitely differentiable, with uni- formly bounded derivatives of all orders, on Rn × [δ, ∞) for each δ 0, we see that u C∞(Rn × (0, ∞)). Furthermore (10) ut(x, t) Δu(x, t) = Rn [(Φt ΔxΦ)(x y, t)]g(y) dy = 0 (x Rn, t 0), since Φ itself solves the heat equation. 2. Fix x0 Rn, ε 0. Choose δ 0 such that (11) |g(y) g(x0)| ε if |y x0| δ, y Rn.
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