This article is devoted to review the known results on global well-posedness for the Cauchy problem to the Kirchhoff equation and Kirchhoff systems with small data. Similar results will be obtained for the initial-boundary value problems in exterior
domains with compact boundary. Also, the known results on large data problems will be reviewed together with open problems.
The aim of this paper is to establish the $H^1$ global well-posedness for Kirchhoff systems. The new approach to the construction of solutions is based on the asymptotic integrations for strictly hyperbolic systems with time-dependent coefficients. T
hese integrations play an important role to setting the subsequent fixed point argument. The existence of solutions for less regular data is discussed, and several examples and applications are presented.
The present paper is devoted to finding a necessary and sufficient condition on the occurence of scattering for the regularly hyperbolic systems with time-dependent coefficients whose time-derivatives are integrable over the real line. More precisely
, it will be shown that the solutions are asymptotically free if the coefficients are stable in the sense of the Riemann integrability as time goes to infinity, while each nontrivial solution is never asymptotically free provided that the coefficients are not R-stable as times goes to infinity. As a by-product, the scattering operator can be constructed. It is expected that the results obtained in the present paper would be brought into the study of the asymptotic behaviour of Kirchhoff systems.
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 asy
mptotic 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.
