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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/ii/paper-4/30c/iv/solution

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Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 4 / 30C / iv / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-14

For the difference of two solutions the equation is

w_{tt} + w_{t} - Δ w = 0

with zero initial data. The same shrinking-ball energy now obeys

E^{'} (t) = - \int_{B_{t}} w_{t}^{2} d x + \int_{\partial B_{t}} [w_{t} \partial_{n} w - \frac{1}{2} (w_{t}^{2} + ∣\nabla w ∣^{2})] d S \leq 0.

(1)

Again

E (0) = 0

forces the difference to vanish at every chosen spacetime point. Therefore the two $C^{2}$ solutions agree for every $t > 0$ . Damping contributes an additional negative energy term and does not weaken the local uniqueness proof.

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