The HADES seminar on Tuesday, February 24th, will be at 3:30pm in Room 740.
Speaker: Hongjing Huang
Abstract:
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, \cite{Delort1997_KG1D} in the semilinear case.
This joint work with Mihaela Ifrim and Daniel Tataru.