Dispersive and Strichartz estimates for hyperbolic equations with constant coefficients

TL;DR

This paper derives dispersive and Strichartz estimates for solutions to high-order strictly hyperbolic PDEs with constant coefficients, analyzing how geometry influences decay rates and applying results to Fokker-Planck and semilinear hyperbolic equations.

Contribution

It provides a comprehensive analysis of decay estimates for high-order hyperbolic equations, linking geometric conditions to decay rates and extending results to specific equations like Fokker-Planck.

Findings

Global $L^p-L^q$ decay estimates are established.

Decay rates depend on geometric properties of the characteristic surfaces.

Applications include decay estimates for Fokker-Planck and semilinear hyperbolic equations.

Abstract

Dispersive and Strichartz estimates for solutions to general strictly hyperbolic partial differential equations with constant coefficients are considered. The global time decay estimates of $L^p-L^q$ norms of propagators are obtained, and it is shown how the time decay rates depend on the geometry of the problem. The frequency space is separated in several zones each giving a certain decay rate. Geometric conditions on characteristics responsible for the particular decay are identified and investigated. Thus, a comprehensive analysis is carried out for strictly hyperbolic equations of high orders with lower order terms of a general form. Results are applied to establish time decay estimates for the Fokker-Planck equation and for semilinear hyperbolic equations.