2.3. HEAT EQUATION 53 A “heat ball” This is a region in space-time, the boundary of which is a level set of Φ(x−y, t−s). Note that the point (x, t) is at the center of the top. E(x, t r) is sometimes called a “heat ball”. THEOREM 3 (A mean-value property for the heat equation). Let u ∈ C1 2(UT ) solve the heat equation. Then (19) u(x, t) = 1 4rn E(x,t r) u(y, s) |x − y|2 (t − s)2 dyds for each E(x, t r) ⊂ UT . Formula (19) is a sort of analogue for the heat equation of the mean-value formulas for Laplace’s equation. Observe that the right-hand side involves only u(y, s) for times s ≤ t. This is reasonable, as the value u(x, t) should not depend upon future times. Proof. Shift the space and time coordinates so that x = 0 and t = 0. Upon mollifying if necessary, we may assume u is smooth. Write E(r) = E(0, 0 r) and set (20) φ(r) := 1 rn E(r) u(y, s) |y|2 s2 dyds = E(1) u(ry, r2s) |y|2 s2 dyds. We compute φ (r) = E(1) n i=1 uyi yi |y|2 s2 + 2rus |y|2 s dyds = 1 rn+1 E(r) n i=1 uyi yi |y|2 s2 + 2us |y|2 s dyds =: A + B.
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