Estimates of oscillatory integrals with stationary phase and singular amplitude: Applications to propagation features for dispersive equations

TL;DR

This paper extends oscillatory integral estimates to include stationary points and singular amplitudes, providing new insights into dispersive wave propagation and localization phenomena for certain equations.

Contribution

It introduces a generalized van der Corput Lemma for phases with stationary points and singular amplitudes, and applies these results to analyze dispersive equations with non-compact initial data.

Findings

Optimal decay rates depend on stationary points and singularities.

Solutions exhibit asymptotic localization in space-time cones.

Symbol growth influences dispersion and causality properties.

Abstract

In this paper, we study time-asymptotic propagation phenomena for a class of dispersive equations on the line by exploiting precise estimates of oscillatory integrals. We propose first an extension of the van der Corput Lemma to the case of phases which may have a stationary point of real order and amplitudes allowed to have an integrable singular point. The resulting estimates provide optimal decay rates which show explicitly the influence of these two particular points. Then we apply these abstract results to solution formulas of a class of dispersive equations on the line defined by Fourier multipliers. Under the hypothesis that the Fourier transform of the initial data has a compact support or an integrable singular point, we derive uniform estimates of the solutions in space-time cones, describing their motions when the time tends to infinity. The method permits also to show that symbols having a restricted growth at infinity may influence the dispersion of the solutions: we prove the existence of a cone, depending only on the symbol, in which the solution is time-asymptotically localized. This corresponds to an asymptotic version of the notion of causality for initial data without compact support.