Invariant Manifolds for a PDE-ODE Coupled System

The aim of this paper is to construct invariant manifolds for a coupled system, consisting of a parabolic equation and a second-order ordinary differential equation, set on \(\mathbb {T}^3\) and subject to periodic boundary conditions. Notably, the “spectral gap condition" does not hold for the system under consideration, leading to the use of the spatial averaging principle, together with the graph transform method. This approach facilitates the construction of the relevant invariant manifold, characterized by attributes such as Lipschitz continuity, local invariance, infinite dimensionality, and exponential tracking, thus mirroring the properties traditionally associated with a classical global manifold.