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 R R and T 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 ϵ persist on the improved cubic timescale |t| ε ⁻² | t | ≲ 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 R R , we are further able to use dispersion in order to extend the lifespan to ε ⁻⁴ ϵ - 4 . This generalizes earlier results obtained by Delort [8] in the semilinear case.