Enhanced lifespan bounds for 1D quasilinear Klein-Gordon flows, (with Mihaela Ifrim and Daniel Tataru), submitted (2026).
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.
A conformal-type energy inequality on hyperboloids and its application to quasi-linear wave equation in 3D, (with Yue Ma), Preprint(2018).