Asymptotically almost periodic solutions to parabolic equations on the real hyperbolic manifold

Abstract

In this work, we study the existence and stability of asymptotically almost periodic mild solutions of vectorial parabolic equations on the real hyperbolic manifold Hd(R) (d⩾2). We will consider the equations associated with Ebin-Marsden's Laplace operator. Our method is based on certain dispersive and smoothing estimates of the semigroup associated with the linearized vectorial heat equation and the fixed point argument. First, we prove the existence and uniqueness of the asymptotically almost periodic mild solution for the linear equation. Next, using the fixed point argument we can pass from the linear equation to the semi-linear equation to prove the existence and uniqueness of such solution. We prove also the exponential stability of solution by using the cone inequality. Finally, abstract results will be applied to Navier-Stokes equations and the semi-linear vectorial heat equation.