Enhanced Lifespan Bounds for 1D Quasilinear Klein–Gordon Flows

Abstract

Abstract In this article we consider one-dimensional scalar quasilinear Klein–Gordon equations with general nonlinearities, on both $${\mathbb {R}}$$ R and $${\mathbb {T}}$$ 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 $$\epsilon $$ ϵ persist on the improved cubic timescale $$|t| \lesssim \epsilon ^{-2}$$ | 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 $${\mathbb {R}}$$ R , we are further able to use dispersion in order to extend the lifespan to $$\epsilon ^{-4}$$ ϵ - 4 . This generalizes earlier results obtained by Delort [8] in the semilinear case.