Sharp time decay estimates for the discrete Klein-Gordon equation

We establish sharp time decay estimates for the Klein–Gordon equation on the cubic lattice in dimensions d = 2, 3, 4. The ℓ1 → ℓ∞ 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...

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Bibliographic Details
Main Authors: Jean-Claude Cuenin, Isroil A Ikromov
Format: Article
Language:English
Subjects:
dispersive and Strichartz estimates
global wellposedness
oscillatory integrals
spectral theory
Online Access:Request full text
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Summary:We establish sharp time decay estimates for the Klein–Gordon equation on the cubic lattice in dimensions d = 2, 3, 4. The ℓ1 → ℓ∞ 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 singularities of the associated phase function. We also prove new Strichartz estimates and discuss applications to nonlinear PDEs and spectral theory.