Asymptotic behavior for quasilinear hyperbolic equations of Kirchhoff type with perturbation having integrable coefficient

Abstract

This paper is concerned with the abstract quasilinear hyperbolic equations of Kirchhoff type with perturbation whose coefficient is integrable in time. We show the unique existence of global solutions for small data in some class and that the solution has the same asymptotic behavior of a function obtained by a transformation of time variable from a solution of the free wave equation with an appropriate wave speed. Conversely, we show that there exists a solution of the Kirchhoff equation which has the same asymptotic behavior of a function obtained by a transformation of the time variable from the solution of the Cauchy problem of the free wave equation with an appropriate wave speed.