On pseudo almost periodic solutions for Boussinesq systems on real hyperbolic manifolds

We study the existence, uniqueness and exponential stability of pseudo almost periodic (PAP-) mild solutions for the Boussinesq systems with the initial data in Lp (Hd (R))-phase space for p > d, where Hd (R) is a real hyperbolic manifold with dimension d ? 2. First, we prove the existence and the uniqueness of the bounded mild solutions for the corresponding linear systems by using dispersive and smoothing estimates of the vectorial matrix semigroup. Then, we prove a Massera-type principle to obtain the existence of PAP-mild solutions to the linear systems. Next, using the fixed point arguments, we can pass from the linear systems to the semilinear systems to establish the well-posedness of such solutions. Finally, we will establish the exponential stability of PAP-mild solutions by using Gronwall?s inequality.