Asymptotic integration and dispersion for hyperbolic equations

TL;DR

This paper develops asymptotic integration methods to analyze decay and dispersion in hyperbolic equations with time-dependent coefficients, providing new insights into their long-term behavior and implications for nonlinear problems.

Contribution

It introduces a novel asymptotic integration approach for hyperbolic equations with time-dependent coefficients and derives explicit decay and dispersive estimates based on geometric indices.

Findings

Decay rates depend on geometric indices of the limiting equation

Representation formulas for solutions are obtained and analyzed

Time decay estimates enable treatment of nonlinear equations with Strichartz estimates

Abstract

The aim of this paper is to establish time decay properties and dispersive estimates for strictly hyperbolic equations with homogeneous symbols and with time-dependent coefficients whose derivatives are integrable. For this purpose, the method of asymptotic integration is developed for such equations and representation formulae for solutions are obtained. These formulae are analysed further to obtain time decay of Lp-Lq norms of propagators for the corresponding Cauchy problems. It turns out that the decay rates can be expressed in terms of certain geometric indices of the limiting equation and we carry out the thorough analysis of this relation. This provides a comprehensive view on asymptotic properties of solutions to time-perturbations of hyperbolic equations with constant coefficients. Moreover, we also obtain the time decay rate of the Lp-Lq estimates for equations of these kinds, so the time well-posedness of the corresponding nonlinear equations with additional semilinearity can be treated by standard Strichartz estimates.