Abstract
This paper is concerned with the large time behavior of the solutions of a parabolic equation on a thin rectangular two-layer domain. The dynamics of the problem on the thin domain is compared with the dynamics of the corresponding limit problem on the reduced domain, which is a line segment. It is shown that provided some regularity condition is satisfied then the initial evolutionary system possesses two modes that uniquely determine its asymptotic dynamics. We also compare our results with those obtained by means of the standard zero-number arguments for scalar ID parabolic equations.
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