Asymptotic equipartition rate for wave motion in an even number of space dimensions (Q1089515)

An energy conserving wave which is initially confined in a sphere of finite radius is propagating in a \(2\ell\)-dimensional space. It is proved that if the Cauchy data have \(\ell +2\) continuous derivatives then the difference between the potential and the kinetic energy has the asymptotic rate of decay \(t^{-2(\ell +\lambda)}\) as time \(t\to +\infty\), where \(\lambda\) depends on the order of the first nonvanishing moment of the Cauchy data. In other words, the rate at which asymptotic equipartition of energy is achieved depends not only on the number of space dimensions but also on how symmetrical the initial disturbance is distributed around the origin.