Enhanced lifespan bounds for 1D quasilinear Klein-Gordon flows
Hongjing Huang, Mihaela Ifrim, Daniel Tataru
TL;DR
This work analyzes long-time dynamics for one-dimensional quasilinear Klein–Gordon equations on $\mathbb{R}$ and $\mathbb{T}$ with mass $m>0$. By combining a refined Ifrim–Tataru modified-energy framework with paradifferential calculus and formal normal forms, it derives cubic-energy estimates that yield enhanced lifespans for small data: on the torus and similar settings the solution exists up to $O(\varepsilon^{-2})$, while on the real line dispersion plus Strichartz extends the lifespan to $O(\varepsilon^{-4})$ under higher regularity. The paper develops a paradifferential reformulation $L^{\mathrm{para}}_{KG}$, constructs energy functionals $E^{s,\mathrm{para}}$ (and the linearized analog $E_{\mathrm{lin}}$), and proves invertibility of normal-form transforms and cubic source bounds for both the full and linearized equations. A bootstrap argument then combines these estimates with Strichartz control to produce the enhanced lifespans and weak-Lipschitz bounds for the solution map, advancing previous semilinear results to the quasilinear regime. The results offer a robust route to long-time well-posedness for 1D quasilinear dispersive flows and highlight the pivotal role of modified energies in managing near-resonant interactions. The techniques have potential applicability to broader quasilinear dispersive systems in low dimensions.
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 $ε$ persist on the improved cubic timescale $|t| \lesssim ε^{-2}$ 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 $ε^{-4}$. This generalizes earlier results obtained by Delort in the semilinear case.