Inertial Manifolds for the Hyperbolic Relaxation of Semilinear Parabolic Equations

TL;DR

This paper develops a new method for constructing inertial manifolds for hyperbolic relaxations of semilinear parabolic equations, establishing spectral gap conditions and analyzing their dependence on relaxation parameters.

Contribution

It introduces a novel scheme for inertial manifold construction and determines optimal spectral gap conditions for their existence in hyperbolic relaxations.

Findings

New construction scheme for inertial manifolds

Optimal spectral gap conditions established

Dependence on relaxation parameter analyzed

Abstract

The paper gives a comprehensive study of Inertial Manifolds for hyperbolic relaxations of an abstract semilinear parabolic equation in a Hilbert space. A new scheme of constructing Inertial Manifolds for such type of problems is suggested and optimal spectral gap conditions which guarantee their existence are established. Moreover, the dependence of the constructed manifolds on the relaxation parameter in the case of the parabolic singular limit is also studied.