Mathematical Sciences - Colloquium

The theory of singularly perturbed differential equations aims to establish the existence of a solution and to determine its asymptotic behavior with respect to the perturbation parameter by means of the corresponding degenerate and associated equation.  The classical approach by N. Tikhonov and N. Levinson and also the geometric theory due to N. Fenichel are based on the crucial assumption that the considered solution of the degenerate equation that represents a family of equilibria of the associated equation does not exhibit an exchange of stability.  This talk represents an extension of the classical approach to the case of exchange of stability where the phenomena of immediate and delayed exchange of stability can be observed.  Our approach is based on the method of asymptotic lower and upper solutions.  We show that this method can be successfully applied to singularly perturbed problems for partial differential equations.

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