Linear and nonlinear tails I: general results and perturbation theory

TL;DR

This paper establishes decay estimates and develops a perturbation theory for nonlinear wave equations with potential, showing that low-order perturbations can accurately predict the decay behavior of solutions.

Contribution

It introduces a novel perturbation approach with a positive convergence radius, reducing wave equations to algebraic forms and providing precise decay insights from first and second orders.

Findings

Perturbation series converges with a positive radius.

First and second order perturbations yield accurate decay predictions.

Method reduces wave equations to algebraic equations for analysis.

Abstract

For nonlinear wave equations with a potential term we prove pointwise space-time decay estimates and develop a perturbation theory for small initial data. We show that the perturbation series has a positive convergence radius by a method which reduces the wave equation to an algebraic one. We demonstrate that already first and second perturbation orders, satisfying linear equations, can provide precise information about the decay of the full solution to the nonlinear wave equation. In a forthcoming publication (part II) we address the issue of optimal decay estimates and precise asymptotics under spherical symmetry where the perturbation equations can be solved almost exactly.