The University of Montana

Department of Mathematical Sciences

Technical report #1/2010

Singularly perturbed periodic parabolic equations with alternating boundary layer type solutions in spatially two-dimensional domains.

**Adelaida B. Vasil'eva**

Department of Physics,

Moscow State University,

Moscow, 119899 Russia

and

** Leonid V. Kalachev**

Department of Mathematical Sciences,

University of Montana,

Missoula, MT 59812, USA

**Abstract**

In this article we continue the analysis of a class of singularly perturbed parabolic equations with alternating boundary layer type solutions. For such problems the degenerate (reduced) equations obtained by setting small parameter equal to zero correspond to algebraic equations that have several isolated roots. As time increases, solutions of these equations periodically go through two comparatively long lasting stages with fast transitions between these stages. During one of these stages the solution outside the boundary layer (i.e., the regular part of the asymptotic solution) is close to one of the roots of the degenerate equation. During the other stage the regular part of the asymptotic solution is close to the other root. Here we discuss some specific features of solutions' behavior for such problems in certain two-dimensional spatial domains.

**Keywords:** singular perturbations, parabolic equations, boundary function method, two-dimensional spatial domains

**AMS Subject Classification:**34E10, 35B05, 35B25

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