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arXiv · 2602.08055

Enhanced lifespan bounds for 1D quasilinear Klein-Gordon flows

Hongjing Huang·Mihaela Ifrim·Daniel Tataru
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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.

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Hongjing Huang, Mihaela Ifrim, Daniel Tataru. 2026-02-08. Enhanced lifespan bounds for 1D quasilinear Klein-Gordon flows. https://arxiv.org/abs/2602.08055

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