Remarks on the existence of an inertial manifold
J. Korean Math. Soc.
Published online November 30, 2020
Minkyu Kwak and Xiuxiu Sun
Chonnam National University
Abstract : 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 a spectral gap condition
is required only for the proof of the uniqueness of such solutions and thus the
uniqueness part 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.
Keywords : Inertial manifolds, asymptotic behaviour of solutions, infinite dimensional dynamical system
MSC numbers : 35B30, 35B40, 35B42
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