46 2. FOUR IMPORTANT LINEAR PDE for r = |y|, = d dr . Now if we set α = n 2 , this simplifies to read (rn−1w ) + 1 2 (rnw) = 0. Thus rn−1w + 1 2 rnw = a for some constant a. Assuming w, w → 0 fast enough as r → ∞, we conclude a = 0 whence w = − 1 2 rw. But then for some constant b (7) w = be− r 2 4 . Combining (4), (7) and our choices for α, β, we conclude that b tn/2 e− |x| 2 4t solves the heat equation (1). This computation motivates the following DEFINITION. The function Φ(x, t) := 1 (4πt)n/2 e− |x|2 4t (x ∈ Rn, t 0) 0 (x ∈ Rn, t 0) is called the fundamental solution of the heat equation. Notice that Φ is singular at the point (0, 0). We will sometimes write Φ(x, t) = Φ(|x|,t) to emphasize that the fundamental solution is radial in the variable x. The choice of the normalizing constant (4π)−n/2 is dictated by the following LEMMA (Integral of fundamental solution). For each time t 0, Rn Φ(x, t) dx = 1. Proof. We calculate Rn Φ(x, t) dx = 1 (4πt)n/2 Rn e− |x| 2 4t dx = 1 πn/2 Rn e−|z|2 dz = 1 πn/2 n i=1 ∞ −∞ e−zi 2 dzi = 1.

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