Abstract and Applied Analysis
Volume 2006 (2006), Article ID 52856, 21 pages

Singularly perturbed periodic parabolic equations with alternating boundary layer type solutions

Adelaida B. Vasil'eva1 and Leonid V. Kalachev2

1Department of Physics, Moscow State University, Moscow 119899, Russia
2Department of Mathematical Sciences, University of Montana, Missoula 59812, MT, USA

Received 30 September 2004; Accepted 4 November 2004

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.


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.