Singularly perturbed boundary value problems for systems of Tichonov's type in case of exchange of stabilities
type: Item
instance of: scholarly article
title: Singularly perturbed boundary value problems for systems of Tichonov's type in case of exchange of stabilities
zbMATH Open document ID: 0944.34048
author: V. F. Butuzov
author: N. N. Nefedov
author: Klaus R. Schneider
published in: Journal of Differential Equations
publication date: 2000-09-11T00:00:00Z
review text: Systems containing ``slow'' and ``fast'' equations are known as Tikhonov-type equations. Here, the authors deal with the ``fast'' equation: \(\varepsilon^2u''= g(u,v,x,\varepsilon)\), where \('\) denotes differentiation with respect to \(x\), and the ``slow'' equation \(v'= f(u,v,x,\varepsilon)\), or a similar equation of second-order. Let \(I_0\) denote the interval \(0<\varepsilon< \varepsilon_0\), with \(\varepsilon_0\ll 1\). \(u\) obeys the no flux across the boundary condition: \(u'(0)= u'(1)= 0\). The variable \(v\) obeys either the initial condition \(v'(0)= v^0\) or the boundary conditions \(v(0)= v^0\), \(v(1)= v^1\). The degenerate equation obtained by setting \(\varepsilon= 0\): \(g(u,v,x,0)= 0\), has two solutions: \(u=\varphi_1(v,x)\) and \(u= \varphi_2(v,x)\) such that \(\varphi_1(v,x)> \varphi_2(v,x)\) for \(v< v^0\), \(\varphi_1(v,x)= \varphi_2(v,x)\) if \(0\leq x\leq 1\), and \(\varphi_1(v,x)> \varphi_2(v,x)\) for \(v> v^0\). Thus this system differs from the original Tikhonov system. In some mathematical circles these types of equations almost automatically bring forth nonstandard analysis arguments; ``Canards'' (duck hunting), infinitely fast motions, and infinitesimal highways seem to appear from nowhere. However, it is a well-known lemma that any results derived by enlarging a space to a nonstandard model can be also derived in the old-fashioned way by taking appropriate limits. In this article, the authors take the classical approach which was pioneered by Chaplygin, Tikhonov, Levinson, O'Malley, and among others, the authors of this article. Here, the authors assume that the two linearly independent solutions may intersect each other. This makes application of most classical techniques originated by Tikhonov not applicable. It is known that such intersecting solutions may interchange their properties, including the stability of solutions. The authors study the stability of degenerate solutions, using the technique of upper and lower solutions …
reviewed by: Vadim Komkov
Mathematics Subject Classification ID: 34E15
Mathematics Subject Classification ID: 34B16
zbMATH DE Number: 1417617
zbMATH Keywords: second-order equations
zbMATH Keywords: singular points
zbMATH Keywords: ``fast''-``slow'' solutions
zbMATH Keywords: Tikhonov's theory
zbMATH Keywords: intersecting solutions
zbMATH Keywords: degenerate solutions
MaRDI profile type: Publication
OpenAlex ID: W2141573592
cites work: A singularly perturbed boundary value problem for a second-order equation in the case of variation of stability
cites work: Q4746140
cites work: Nonlinear singular perturbation phenomena: theory and applications
cites work: Subfunction and second-order ordinary differential inequalities
cites work: Exchange of Stabilities in Autonomous Systems
cites work: Q5839062
cites work: Q4370405
cites work: Immediate exchange of stabilities in singularly perturbed systems
cites work: Nonlinear elliptic boundary-value problems in unbounded domains
cites work: Q5814121
DOI: 10.1006/JDEQ.1999.3670
description: scientific article; zbMATH DE number 1417617