Global Well-posedness for 2D Nonlinear Wave Equations without Compact Support

TL;DR

This paper proves global well-posedness for small solutions to 2D nonlinear wave equations under null conditions without requiring initial data to have compact support, extending previous results.

Contribution

It establishes the first global existence result for fully nonlinear 2D wave equations without the compact support assumption.

Findings

Global well-posedness for small initial data without compact support

Application to a class of quasilinear wave equations

Extension of null condition results to broader initial data settings

Abstract

In the significant work of [2], Alinhac proved the global existence of small solutions for 2D quasilinear wave equations under the null conditions. The proof heavily relies on the fact that the initial data have compact support [22]. Whether this constraint can be removed or not is still unclear. In this paper, for fully nonlinear wave equations under the null conditions, we prove the global well-posedness for small initial data without compact support. Moreover, we apply our result to a class of quasilinear wave equations.