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

TL;DR

This paper investigates the existence, uniqueness, and asymptotic behavior of almost periodic solutions to vectorial parabolic equations on hyperbolic manifolds, with applications to Navier-Stokes and heat equations.

Contribution

It introduces a method using dispersive estimates and fixed point arguments to establish asymptotically almost periodic solutions on hyperbolic manifolds, extending previous results to vectorial and nonlinear cases.

Findings

Proved existence and uniqueness of solutions for linearized equations.

Established exponential decay and stability of solutions.

Applied results to Navier-Stokes and vectorial heat equations.

Abstract

In this work we study the existence and the asymptotic behaviour of the asymptotically almost periodic mild solutions of the vectorial parabolic equations on the real hyperbolic manifold $\mathbb{H}^d(\mathbb{R})$ ($d \geqslant 2$). We will consider the vectorial laplace operator in the sense of Ebin-Marsden's laplace operator. Our method is based on certain dispertive and smoothing estimates of the semigroup generated by the linearized vectorial heat equation and the fixed point argument. First, we prove the existence and the uniqueness of the asymptotically almost periodic mild solution for the linearized equations. Next, using the fixed point argument, we can pass from linearized equations to semilinear equations to prove the existence, uniqueness, exponential decay and stability of the solutions. Our abstract results will be applied to the incompressible Navier-Stokes equation and the semilinear vectorial heat equation.