# Global stability for nonlinear wave equations with multi-localized initial data

## TL;DR

This paper establishes the global stability of nonlinear wave equations with multi-localized initial data, extending previous results to configurations where data points are arbitrarily far apart and not necessarily localized around a single point.

## Contribution

It introduces a novel approach analyzing wave interactions from multiple sources and proves global stability without scale-dependent smallness conditions.

## Key findings

- Proves global existence for data with large $H^1$ norm.
- Develops bilinear estimates for wave interactions from multiple sources.
- Extends stability results to non-radially localized initial data.

## Abstract

In this paper, we initiate the study of the global stability of nonlinear wave equations with initial data that are not required to be localized around a single point. More precisely, we allow small initial data localized around any finite collection of points which can be arbitrarily far from one another. Existing techniques do not directly apply to this setting because they require norms with radial weights away from some center to be small. The smallness we require on the data is measured in a norm which does not depend on the scale of the configuration of the data. Our method of proof relies on a close analysis of the geometry of the interaction between waves originating from different sources. We prove estimates on the bilinear forms encoding the interaction, which allow us to show improved bounds for the energy of the solution. We finally apply a variant of the vector field method involving modified Klainerman--Sobolev estimates to prove global stability. As a corollary of our proof, we are able to show global existence for a class of data whose $H^1$ norm is arbitrarily large.