Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅱ: periodic boundary conditions

Anna Kostianko; Sergey Zelik

doi:10.3934/cpaa.2018017

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2018, Volume 17, Issue 1: 285-317. Doi: 10.3934/cpaa.2018017

This issue Previous Article On stability of functional differential equations with rapidly oscillating coefficients Next Article

Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅱ: periodic boundary conditions

Anna Kostianko and
Sergey Zelik^,

University of Surrey, Department of Mathematics, Guildford, GU2 7XH, United Kingdom

* Corresponding author
* Corresponding author

Received: February 28, 2017

Published: December 2017

This work is partially supported by the grants 14-41-00044 and 14-21-00025 of RSF as well as grants 14-01-00346 and 15-01-03587 of RFBR.

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Abstract

This is the second part of our study of the Inertial Manifolds for 1D systems of reaction-diffusion-advection equations initiated in [6] and it is devoted to the case of periodic boundary conditions. It is shown that, in contrast to the case of Dirichlet or Neumann boundary conditions, considered in the first part, Inertial Manifolds may not exist in the case of systems endowed with periodic boundary conditions. However, as also shown, inertial manifolds still exist in the case of scalar reaction-diffusion-advection equations. Thus, the existence or non-existence of inertial manifolds for this class of dissipative systems strongly depend on the choice of boundary conditions.

Keywords:

Inertial manifolds,
convective reaction-diffusion equation.

Mathematics Subject Classification: Primary:35B40, 35B45.

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