Eventual partition of conserved quantities in wave motion

Let 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 $\text{(n + 2)(n + 3)}\text{2}$ 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.