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 Incoming links: Outgoing shell source estimate for a radial wave

Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 10 / 1 / 4 / Solution 2026-10-06

Use the radial reduction of the three-dimensional wave equation

w = r ϕ

and extend

w

oddly in

r

. Write

w = w_{h} + w_{F}

, where

w_{h}

is the homogeneous wave equation solution with the prescribed Cauchy data, and

w_{F}

has zero Cauchy data. With

w_{0} (r) = r ϕ_{0} (r)

and

w_{1} (r) = r ϕ_{1} (r)

, the D'Alembert formula gives

w_{h} (t, r) = \frac{w _{0} ( r - t ) + w _{0} ( r + t )}{2} + \frac{1}{2} \int_{r - t}^{r + t} w_{1} (ρ) d ρ .

Both odd initial profiles have compact support, so

∣ w_{h} (t, r) ∣ \leq A, A = ∥ w_{0} ∥_{\infty} + \frac{1}{2} ∥ w_{1} ∥_{1} .

The Duhamel principle, applied to

w_{tt} - w_{rr} = - r F

, gives

w_{F} (t, r) = - \frac{1}{2} \int_{0}^{t} \int_{r - (t - s)}^{r + (t - s)} ρF (s, ρ) d ρ d s .

Now write

r = t + a

with

1/2 \leq a \leq 1

. The lower integration limit is

s + a > 0

, so this domain of dependence never crosses the axis. Intersecting its integration interval with the support of the source leaves a subset of

[s + a, s + 1]

, of length at most one. On this interval,

ρ ∣ F (s, ρ) ∣ \leq \frac{s + 1}{( 1 + s ) ^{2}} = \frac{1}{1 + s} .

The outgoing shell source estimate for a radial wave therefore yields

∣ w_{F} (t, t + a) ∣ \leq \frac{1}{2} \int_{0}^{t} \frac{d s}{1 + s} = \frac{1}{2} lo g (1 + t) .

Finally

r \geq t + 1/2 \geq (1 + t) /2

, and thus

∣ ϕ (t, t + a) ∣ \leq \frac{2 A + l o g ( 1 + t )}{1 + t} \leq \frac{C lo g ( 2 + t )}{1 + t}, C = 1 + \frac{2 A}{l o g 2} .

This constant depends only on the Cauchy data and the fixed unit bound for the source. The logarithm comes from integrating the shell's accumulated forcing, whereas the factor

(1 + t)^{- 1}

comes from division by

r

.

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