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Decay of solutions of symmetric hyperbolic systems of partial differential equations

We consider systems of the form ut + 2 ^ A-ux — 0, where the A/s are constant k X k (hermitian) symmetric matrices, and u is a column vector of k components. We use Fourier transform to prove that non-static solutions decay in time at every point x. As a consequence, it follows that the energy of any such solution decays locally. More generally, we show that if B(t) is a set which does not increase "too" fast, the energy in B(t) of any non-static solution also decays.