A note on late-time tails of spherical nonlinear waves

We consider the long-time behavior of small amplitude solutions of the semilinear wave equation $\Box \phi =\phi^p$ in odd $d\geq 5$ spatial dimensions. We show that for the quadratic nonlinearity ($p=2$) the tail has an anomalously small amplitude and fast decay. The extension of the results to more general nonlinearities involving first derivatives is also discussed.