2.3. HEAT EQUATION 49 is in fact positive for all points x ∈ Rn and times t 0. We interpret this observation by saying the heat equation forces infinite propagation speed for disturbances. If the initial temperature is nonnegative and is positive somewhere, the temperature at any later time (no matter how small) is everywhere positive. (We will learn in §2.4.3 that the wave equation in contrast supports finite propagation speed for disturbances.) c. Nonhomogeneous problem. Now let us turn our attention to the nonhomogeneous initial-value problem (12) ut − Δu = f in Rn × (0, ∞) u = 0 on Rn × {t = 0}. How can we produce a formula for the solution? If we recall the moti- vation leading up to (9), we should note further that the mapping (x, t) → Φ(x−y, t−s) is a solution of the heat equation (for given y ∈ Rn, 0 s t). Now for fixed s, the function u = u(x, t s) = Rn Φ(x − y, t − s)f(y, s) dy solves (12s) ut(· s) − Δu(· s) = 0 in Rn × (s, ∞) u(· s) = f(·,s) on Rn × {t = s}, which is just an initial-value problem of the form (8), with the starting time t = 0 replaced by t = s and g replaced by f(·,s). Thus u(· s) is certainly not a solution of (12). However Duhamel’s principle∗ asserts that we can build a solution of (12) out of the solutions of (12s), by integrating with respect to s. The idea is to consider u(x, t) = t 0 u(x, t s) ds (x ∈ Rn, t ≥ 0). Rewriting, we have (13) u(x, t) = t 0 Rn Φ(x − y, t − s)f(y, s) dyds = t 0 1 (4π(t − s))n/2 Rn e− |x−y| 2 4(t−s) f(y, s) dyds, for x ∈ Rn, t 0. To confirm that formula (13) works, let us for simplicity assume f ∈ C1 2(Rn × [0, ∞)) and f has compact support. ∗Duhamel’s principle has wide applicability to linear ODE and PDE and does not depend on the specific structure of the heat equation. It yields, for example, the solution of the nonho- mogeneous transport equation, obtained by different means in §2.1.2. We will invoke Duhamel’s principle for the wave equation in §2.4.2.

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