Sharp time decay estimates for the discrete Klein–Gordon equation

Abstract We establish sharp time decay estimates for the Klein–Gordon equation on the cubic lattice in dimensions d = 2, 3, 4. The ℓ 1 → 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.

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