Enhanced lifespan bounds for 1D quasilinear Klein-Gordon flows

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.