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

Abstract

We study the existence, uniqueness and exponential stability of pseudo almost periodic (PAP-) mild solutions for the Boussinesq systems with the initial data in $L^p(\mathbb{H}^d(\mathbb{R}))$-phase space for $p>d$, where $\mathbb{H}^d(\mathbb{R})$ is a real hyperbolic manifold with dimension $d \geqslant 2$. 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-mildsolutions 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.