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Abstract
In this paper we are concerned with a wave equation with damping in which one of the boundary conditions is of a dynamic nature. The problem serves to describe the longitudinal vibrations of a homogeneous flexible horizontal rod in which one end is rigidly fixed while the other end is free to move with an attached load. Viscous effects are present both within the rod and at that end of the rod to which the load is attached, with the effect that damping of Kelvin-Voigt type occurs in both the partial differential equation and the dynamic boundary condition. The problem is studied within the framework of the abstract theories of B-evolutions and fractional powers of a closed pair of operators by formulating an abstract evolution problem in the product space X ½ x X with X a Hilbert space and X ½ the domain of a fractional power of a closed pair of operators with domain D in X.
