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

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.