In this paper the existence, uniqueness and regularity properties to solution of the Cauchy problem for the fractional abstract Boussinesq equation is obtained. First, we consider the linear Boussinesq equation and prove the existence, uniqueness and the regularity properties of solutions. It can be used to obtain the existence and uniqueness of the regular solution of corresponding nonlinear Boussinesq equation. By applying this result, the Cauchy problem for finite or infinite systems of Boussinesq equations is studied.