A note on late-time tails of spherical nonlinear waves

TL;DR

This paper investigates the late-time decay of small solutions to semilinear wave equations in higher odd dimensions, revealing an anomalously small amplitude tail for quadratic nonlinearities and discussing extensions to more general cases.

Contribution

It provides new insights into the decay behavior of nonlinear wave solutions in higher dimensions, especially highlighting the anomalous tail for quadratic nonlinearities.

Findings

Quadratic nonlinearities produce an anomalously small amplitude tail.

Solutions decay faster than expected in odd dimensions ≥ 5.

Extensions to nonlinearities with first derivatives are discussed.

Abstract

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.