  The initial- and boundary-value problem for the Benjamin-Bona-Mahony (BBM)
equation is studied in this paper. The goal is to understand the periodic
behavior (termed as eventual periodicity) of its solutions corresponding to
periodic boundary condition or periodic forcing. To this aim, we derive a new
formula representing solutions of this initial- and boundary-value problem by
inverting the operator $\partial_t +\alpha \partial_x -\gamma\partial_{xxt}$
defined in the space-time quarter plane. The eventual periodicity of the
linearized BBM equation with periodic boundary data and forcing term is
established by combining this new representation formula and the method of
stationary phase.
