Eventual partition of conserved quantities in wave motion

AbstractLet u be a classical solution to the wave equation in an odd number n of space dimensions, with compact spatial support at each fixed time. Duffin (J. Math. Anal. Appl.32 (1970), 386–391) uses the Paley-Wiener theorem of Fourier analysis to show that, after a finite time, the (conserved) energy of u is partitioned into equal kinetic and potential parts. The wave equation actually has (n + 2)(n + 3)2 independent conserved quantities, one for each of the standard generators of the conformal group of (n + 1)-dimensional Minkowski space. Of concern in this paper is the “zeroth inversional quantity” I0, which is commonly used to improve decay estimates which are obtained using conservation of energy. We use Duffin's method to partition I0 into seven terms, each of which, after a finite time, is explicitly given as a constant-coefficient quadratic function of the time. Zachmanoglou has shown that under the above assumptions if n ⩾ 3, the spatial L2 norm of u is eventually constant. A consequence of the analysis here is a bound on this constant in terms of the energy and the radius of the support of the Cauchy data of u at a fixed time