Singularly perturbed periodic parabolic equations with alternating boundary layer type solutions

Vasil'eva, Adelaida B.; Kalachev, Leonid V.

doi:https://doi.org/10.1155/AAA/2006/52856

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Volume 2006 | Article ID 052856 | https://doi.org/10.1155/AAA/2006/52856

Singularly perturbed periodic parabolic equations with alternating boundary layer type solutions

Adelaida B. Vasil'eva¹and Leonid V. Kalachev²

Received30 Sept 2004

Accepted04 Nov 2004

Published21 Feb 2006

Abstract

We consider a class of singularly perturbed parabolic equations for which the degenerate equations obtained by setting the small parameter equal to zero are algebraic equations that have several roots. We study boundary layer type solutions that, as time increases, periodically go through two fairly long lasting stages with extremely fast transitions in between. During one of these stages the solution outside the boundary layer is close to one of the roots of the degenerate (reduced) equation, while during the other stage the solution is close to the other root. Such equations may be used as models for bio-switches where the transitions between various stationary states of biological systems are initiated by comparatively slow changes within the systems.

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Copyright

Copyright © 2006 Adelaida B. Vasil'eva and Leonid V. Kalachev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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