Journal of the
Korean Mathematical Society JKMS

An inertial manifold is often constructed as a graph of a function from low Fourier modes to high ones and one used to consider backward bounded (in time) solutions for that purpose. We here show that the proof of the uniqueness of such solutions is crucial in the existence theory of inertial manifolds. Avoiding contraction principle, we mainly apply the Arzela-Ascoli theorem and Laplace transform to prove their existence and uniqueness respectively. A non-self adjoint example is included, which is related to a differential system arising after Kwak transform for Navier-Stokes equations.

Keywords: Inertial manifolds, asymptotic behaviour of solutions, infinite dimensional dynamical system

MSC numbers: 35B30, 35B40, 35B42

Supported by: This work was supported by Basic Science Research Program through NRF, 2017R1E1A1A03070061.