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.
Highlights
Introduction and main resultsDispersive estimates play a crucial role in the study of evolution equations
In the discrete case the frequencies are bounded, and one might expect the same decay rate for the discrete Klein–Gordon equation (DKG) as for the discrete Schrödinger equation (DS). This was conjectured by Kevrekidis and Stefanov [18], who proved the sharp |t|−d/3 decay rate for the DS in any dimension d 1 and for the DKG in one dimension
Unbeknownst to the authors, earlier, Schultz [26] had already made the striking observation that the decay rate for the discrete wave equation in dimensions d = 2, 3 is |t|−3/4 and |t|−7/6, respectively, better than the conjectured estimates
Summary
Dispersive estimates play a crucial role in the study of evolution equations. Proving such estimates often boils down to establishing decay estimates for the ∞ norm of the solution at time t in terms of the 1 norm of its initial data. The proof of theorem 1 gives more precise information on the set of velocities for which the indicated decay occurs These velocities are the images, under the map ξ → ∇ω(ξ), of the critical points of the phase function Φ(·, v). X1k+1 x21 x2 + xk− x41 + x42 + x43 + ax x2 x3, a = 0 k−1 2k+2 k−2 2k−2 phase function has only critical points with finite multiplicity, and in the most degenerate case is similar to a hyperbolic singularity (T4,4,4 in the classification of Arnol’d [1, 15.1]). The critical point of a hyperbolic singularity is (complex) isolated; our uniform estimates hold true for more general phase functions having non-isolated critical points.
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