Global existence and asymptotics for quasi-linear one-dimensional Klein-Gordon equations with mildly decaying Cauchy data

TL;DR

This paper proves global existence and describes the long-term behavior of solutions to a one-dimensional quasi-linear Klein-Gordon equation with mildly decaying initial data, extending previous results to non-compactly supported data.

Contribution

It extends global existence results to solutions with non-compactly supported, mildly decaying data using combined analytical methods and provides a detailed asymptotic expansion for large times.

Findings

Solutions exist globally for data decaying as  at infinity.

Established a one-term asymptotic expansion for solutions as time approaches infinity.

Combined Klainerman vector fields with semiclassical normal forms for analysis.

Abstract

Let u be a solution to a quasi-linear Klein-Gordon equation in one-space dimension, $\Box u + u = P (u, $\partial$\_t u, $\partial$\_x u; $\partial$\_t $\partial$\_x u, $\partial$^2\_x u)$ , where P is a homogeneous polynomial of degree three, and with smooth Cauchy data of size $\epsilon \rightarrow 0$. It is known that, under a suitable condition on the nonlinearity, the solution is global-in-time for compactly supported Cauchy data. We prove in this paper that the result holds even when data are not compactly supported but just decaying as $\langle x \rangle^ {--1}$ at infinity, combining the method of Klainerman vector fields with a semiclassical normal forms method introduced by Delort. Moreover, we get a one term asymptotic expansion for u when $t \rightarrow +\infty$.