Asymptotic integration and dispersion for hyperbolic equations

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 belong to $L^1(\mathbb{R})$. 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 $L^p$--$L^q$ 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 a thorough analysis of this relation. This provides a comprehensive view of asymptotic properties of solutions to time-perturbations of hyperbolic equations with constant coefficients. Moreover, we also obtain the time decay rate of the $L^p$--$L^q$ estimates for equations of these kinds, so that the time well posedness of the corresponding nonlinear equations with additional semilinearity can be treated by standard Strichartz estimates.