N. N. Nefedov, L. Recke, K. R. Schneider, “Asymptotic stability via the Krein–Rutman theorem for singularly perturbed parabolic periodic Dirichlet problems”, Regul. Chaotic Dyn., 2010, Volume 15, Issue 2-3,Pages <nobr>382

This article is cited in 7 papers

On the 75th birthday of Professor L.P. Shilnikov

Asymptotic stability via the Krein–Rutman theorem for singularly perturbed parabolic periodic Dirichlet problems

N. N. Nefedov^a, L. Recke^b, K. R. Schneider^c

^a Faculty of Physics, M.V. Lomonosov Moscow State University, Leninskie Gory, Moscow, 119991 Russia
^b Humboldt-Universität zu Berlin, Institut für Mathematik, Unter den Linden 6, D-10099 Berlin, Germany
^c Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany

Abstract: We consider singularly perturbed semilinear parabolic periodic problems and assume the existence of a family of solutions. We present an approach to establish the exponential asymptotic stability of these solutions by means of a special class of lower and upper solutions. The proof is based on a corollary of the Krein–Rutman theorem.

Keywords: singularly perturbed parabolic Dirichlet problems, exponential asymptotic stability, Krein–Rutman theorem, lower and upper solutions.

MSC: 35B10, 35B25, 35B35, 35K58, 35K90

Received: 05.11.2009
Accepted: 12.12.2009

Language: English

DOI: 10.1134/S1560354710020231