Sharp time decay estimates for the discrete Klein–Gordon equation
Jean‐Claude CueninDepartment of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, United KingdomIsroil A. IkromovInstitute of Mathematics named after V.I. Romanovsky Academy of Sciences of Uzbekistan, University Boulevard, 15, 140104, Samarkand, Uzbekistan
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Аннотация
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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