4.3 Energy
Theorem 4.3.1 (Conservation of energy). Let \(u\) be a \(C^2\) solution of (4.1) for which \(u_t\) and \(u_x\) vanish for large \(|x|\). Then the energy \[ E(t) = \frac 12\int _{-\infty }^\infty \bigl (u_t^2 + c^2u_x^2\bigr )\,dx \] is constant. More generally, for any \(a < b\) the energy in the shrinking interval \([a + ct, b - ct]\) is non-increasing for \(0 \leq t \leq (b - a)/2c\).
Proof. Let \(e = \frac 12(u_t^2 + c^2u_x^2)\). Then \(e_t = u_tu_{tt} + c^2u_xu_{xt} = c^2(u_tu_{xx} + u_xu_{xt}) = c^2(u_tu_x)_x\), so \(dE/dt = c^2[u_tu_x]_{-\infty }^\infty = 0\). For the shrinking interval, by Leibniz’s rule, \[ \frac {d}{dt}\int _{a + ct}^{b - ct}e\,dx = c^2\bigl [u_tu_x\bigr ]_{a + ct}^{b - ct} - c\,e(b - ct) - c\,e(a + ct) , \] and \(|c^2u_tu_x| \leq c\cdot \frac 12(u_t^2 + c^2u_x^2) = c\,e\), so each endpoint contributes at most \(0\). □
The second statement gives uniqueness and finite speed of propagation without any formula. If two solutions agree initially on \([a, b]\), their difference has zero energy there at \(t = 0\), hence zero energy on the shrinking interval, hence \(u_t = u_x = 0\) and the difference vanishes in the triangle: this is the domain of dependence again. The argument works for variable coefficients and in any dimension, where no d’Alembert formula exists, and it is the model for the energy methods of a later section.
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