Enhanced lifespan bounds for 1D quasilinear Klein-Gordon flows

TL;DR

This paper improves lifespan bounds for small solutions to 1D quasilinear Klein-Gordon equations, extending the lifespan from cubic to quartic timescales using refined energy methods and dispersion analysis.

Contribution

It introduces a refined modified-energy framework to establish longer lifespan bounds for small data solutions in 1D quasilinear Klein-Gordon equations, extending previous semilinear results.

Findings

Solutions persist on cubic timescale with sharp energy estimates

Dispersion extends lifespan to quartic timescale on 

Generalizes earlier semilinear lifespan results

Abstract

In this article we consider one-dimensional scalar quasilinear Klein--Gordon equations with general nonlinearities, on both $\mathbb{R}$ and $\mathbb{T}$. By employing a refined modified-energy framework of Ifrim and Tataru, we investigate long time lifespan bounds for small data solutions. Our main result asserts that solutions with small initial data of size $\epsilon$ persist on the improved cubic timescale $|t| \lesssim \epsilon^{-2}$ and satisfy sharp cubic energy estimates throughout this interval. We also establish difference bounds on the same time scale. In the case of $\mathbb{R}$, we are further able to use dispersion in order to extend the lifespan to $\epsilon^{-4}$. This generalizes earlier results obtained by Delort in the semilinear case.