Dynamics of Partial Differential Equations

Volume 13 (2016)

Number 3

On the hyperbolicity properties of inertial manifolds of reaction–diffusion equations

Pages: 263 – 272

DOI: http://dx.doi.org/10.4310/DPDE.2016.v13.n3.a4

Author

A. V. Romanov (Higher School of Economics, National Research University, Moscow, Russia)

Abstract

For 3D reaction–diffusion equations, we study the problem of existence or nonexistence of an inertial manifold that is normally hyperbolic or absolutely normally hyperbolic. We present a system of two coupled equations with a cubic nonlinearity which does not admit a normally hyperbolic inertial manifold. An example separating the classes of such equations admitting an inertial manifold and a normally hyperbolic inertial manifold is constructed. Similar questions concerning absolutely normally hyperbolic inertial manifolds are discussed.

Keywords

reaction–diffusion equations, inertial manifold, normal hyperbolicity

2010 Mathematics Subject Classification

Primary 35B42, 35K57. Secondary 35K90, 35K91.

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Published 23 June 2016