Alternating boundary layer type solutions of some singularly perturbed periodic parabolic equations with Dirichlet and Robin boundary conditions

Abstract

In this paper, we continue the analysis of alternating boundary layer type solutions to certain singularly perturbed parabolic equations for which the degenerate equations (obtained by setting small parameter multiplying derivatives equal to zero) are algebraic equations that have three roots. Here, we consider spatially one-dimensional equations. We address special cases where the following are true: (a) boundary conditions are of the Dirichlet type with different values of unknown functions specified at different endpoints of the interval of interest; (b) boundary conditions are of the Robin type with an appropriate power of a small parameter multiplying the derivative in the conditions. We emphasize a number of new features of alternating boundary layer type solutions that appear in these cases. One of the important applications of such equations is related to modeling certain types of bioswitches. Special choices of Dirichlet and Robin type boundary conditions can be used to tune up such bioswitches.