Abstract and Applied Analysis
Volume 2006 (2006), Article ID 52856, 21 pages
doi:10.1155/AAA/2006/52856
Singularly perturbed periodic parabolic equations with alternating boundary layer type solutions
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.
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.