Exponential and sub-exponential stability times for the derivative wave equation

Abstract

We prove a Nekhoroshev type theorem for the derivative wave equation$ \begin{equation*} u_{tt} = u_{xx}-mu-(D^{\alpha}u)^3, \quad \ D = \sqrt{-\partial_{xx}+m} \end{equation*} $under Dirichlet boundary conditions and $ 0\leq \alpha\leq 1 $. More precisely, we prove the sub-exponential long time stability result in a smooth function space and the exponential long time stability result in a modified Sobolev space around the origin for the above equation, where using Birkhoff normal form technique for infinite dimensional Hamiltonian systems and the so-called $ {tame} $ property of the non-linearity. This result is inspired by Bambusi-Grébert (2006, Duke) and a recent work by Feola-Massetti (arXiv: 2207.09986v1).