Absence and Blow-Up of Solutions to Singular Perturbation Problems in the Case of Exchange of Stabilities

Abstract

We consider some singular perturbation problems in the case where a degenerate equation has intersecting roots (this case is also referred to as ‘ the exchange of stabilities’). Such problems often occur as models in chemical kinetics. There are lots of works that establish the existence and asymptotic behavior of solutions to such problems. Due to exchange of stabilities, a typical solution approaches the non-smooth (but continuous) composite root of the degenerate equation as the perturbation parameter gets smaller. In a number of problems a regular part of the perturbative term dominates the singular one, so an additional condition on the regular part is needed to improve the stability of a composite root in the vicinity of the intersection point. Inversion of that condition results in a loss or a blow-up of the solution for sufficiently small values of the perturbation parameter. We prove some results of this kind by means of the nonlinear capacity argument and discuss their role in developing numerical algorithms for the problems under consideration.