On the Local Existence of Solutions to an Abstract Quasilinear Second Order Kirchhoff Equation with a Nonlinear Inhomogeneous Term and Internal Viscous Damping of Fractional Type.

Abstract

The purpose of this paper is the study of the local existence of solutions to an abstract quasi-linear Kirchhoff equation with a nonlinear in-homogeneous term submitted to an internal viscous damping of fractional type. We establish the local existence using a method introduced by Kato{\bf\cite{Kato1}} combined with the multiplier method and some fixed point argument.