
Mathematics 2007
Asymptotic integration and dispersion for hyperbolic equationsAbstract: The aim of this paper is to establish time decay properties and dispersive estimates for strictly hyperbolic equations with homogeneous symbols and with timedependent 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 LpLq 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 timeperturbations of hyperbolic equations with constant coefficients. Moreover, we also obtain the time decay rate of the LpLq estimates for equations of these kinds, so the time wellposedness of the corresponding nonlinear equations with additional semilinearity can be treated by standard Strichartz estimates.
