Ground states of singularly perturbed convection-diffusion equation with oscillating coefficients

¹ Narvik University College, Postboks 385, 8505 Narvik, Norway, and P.N. Lebedev Physical Institute of RAS, 53, Leninski pr., 119991 Moscow, Russia
andrey@sci.lebedev.ru
² Kharkiv National University of Economics, 9a Lenin ave., 61166 Kharkiv, Ukraine
nrybalko@yahoo.com
³ Mathematical Department, B.Verkin Institute for Low Temperature Physics and Engineering of the NASU, 47 Lenin ave., 61103 Kharkiv, Ukraine,
vrybalko@ilt.kharkov.ua

Received: 19 March 2013
Revised: 16 November 2013

Abstract

We study the first eigenpair of a Dirichlet spectral problem for singularly perturbed convection-diffusion operators with oscillating locally periodic coefficients. It follows from the results of [A. Piatnitski and V. Rybalko, On the first eigenpair of singularly perturbed operators with oscillating coefficients. Preprint www.arxiv.org, arXiv:1206.3754] that the first eigenvalue remains bounded only if the integral curves of the so-called effective drift have a nonempty ω-limit set. Here we consider the case when the integral curves can have both hyperbolic fixed points and hyperbolic limit cycles. One of the main goals of this work is to determine a fixed point or a limit cycle responsible for the first eigenpair asymptotics. Here we focus on the case of limit cycles that was left open in [A. Piatnitski and V. Rybalko, Preprint.