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

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 ${\mathbb{H}}^d(\mathbb{R})$ ($d⩾2$). 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.

20 pages. arXiv admin note: text overlap with arXiv:2101.03306

References in corpus (1)

• The incompressible Navier-Stokes equations on non-compact manifolds

Cited by in corpus (5)

• On asymptotically almost periodic solutions to the Navier-Stokes equations on hyperbolic manifolds
• Asymptotically almost Periodic Solutions of the parabolic-parabolic Keller-Segel systems on bounded domains
• Generalized gravitational fields and well-posedness of the Boussinesq systems on non-compact Riemannian Manifolds
• On pseudo almost Periodic Solutions of the parabolic-elliptic Keller-Segel systems
• Well-posedness and exponential stability for Boussinesq systems on real hyperbolic Manifolds and application