Sharp time decay estimates for the discrete Klein-Gordon equation

Abstract

We establish sharp time decay estimates for the the Klein-Gordon equation on the cubic lattice in dimensions $d=2,3,4$. The ${\ell }^1\to {\ell }^{\infty }$ dispersive decay rate is $|t{|}^{-3/4}$ for $d=2$, $|t{|}^{-7/6}$ for $d=3$ and $|t{|}^{-3/2}\log |t|$ for $d=4$. These decay rates are faster than conjectured by Kevrekidis and Stefanov (2005). The proof relies on oscillatory integral estimates and proceeds by a detailed analysis of the the singularities of the associated phase function. We also prove new Strichartz estimates and discuss applications to nonlinear PDEs and spectral theory.

exposition improved, some tyops corrected

References in corpus (4)

• Excitation Thresholds for Nonlinear Localized Modes on Lattices
• Asymptotic behavior of small solutions for the discrete nonlinear Schrödinger and Klein-Gordon equations
• The Klein-Gordon Equation on ${\mathbb{Z}}^2$ and the Quantum Harmonic Lattice
• Mixed norm Strichartz-type estimates for hypersurfaces in three dimensions