Global pointwise decay estimates for defocusing radial nonlinear wave equations

TL;DR

This paper establishes optimal global pointwise decay estimates for spherically symmetric defocusing nonlinear wave equations in three dimensions using conformal transformations, applicable to large initial data.

Contribution

It introduces a novel approach employing conformal transformations tailored to the nonlinearity to derive decay estimates for large data solutions.

Findings

Decay rates match those for small data, indicating optimality.

Applicable to arbitrarily large initial data if solutions are global.

Method suggests potential generalization beyond spherical symmetry.

Abstract

We prove global pointwise decay estimates for a class of defocusing semilinear wave equations in $n=3$ dimensions restricted to spherical symmetry. The technique is based on a conformal transformation and a suitable choice of the mapping adjusted to the nonlinearity. As a result we obtain a pointwise bound on the solutions for arbitrarily large Cauchy data, provided the solutions exist globally. The decay rates are identical with those for small data and hence seem to be optimal. A generalization beyond the spherical symmetry is suggested.