Estimates for a class of oscillatory integrals and decay rates for wave-type equations

Abstract

This paper investigates higher order wave-type equations of the form ∂ttu+P(Dx)u=0, where the symbol P(ξ) is a real, non-degenerate elliptic polynomial of the order m≥4 on boldRn. Using methods from harmonic analysis, we first establish global pointwise time–space estimates for a class of oscillatory integrals that appear as the fundamental solutions to the Cauchy problem of such wave equations. These estimates are then used to establish (pointwise-in-time) Lp−Lq estimates on the wave solution in terms of the initial conditions.

On the nonlinear Dysthe equation

“…The following results were presented in the second author's PhD thesis [27]. Similar results have been obtained by Ben-Artzi-Koch-Saut for a general class of third-order dispersive equations in 2D [5] (see also [1,11,35,36]). In fact their results yield an overall decay of |t| −2/3 in R 2 .…”

“…Based on the work of Kenig, Ponce and Vega on the KdV equation [24], one expects to derive linear smoothing estimates in a space such as L ∞ x L 2 t,y . A key ingredient in the proof, however, would be making sure that 1 There is a Hamiltonian version of the 1D Dysthe equation [9], but the existence of a Hamiltonian version in 2D remains an open question. 2 In this regard it is important to note that in [14] the authors argue that their goal is not longtime prediction.…”