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Complete Asymptotics for Solution of Singularly Perturbed Dynamical Systems with Single Well Potential

Author

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  • Denis I. Borisov

    (Department of Differential Equations, Institute of Mathematics, Ufa Federal Research Center, Russian Academy of Sciences, Chernyshevsky str. 112, 450008 Ufa, Russia
    Faculty of Mathematics, Bashkir State University, Zaki Validi str. 32, 450000 Ufa, Russia
    Faculty of Science, University of Hradec Králové, Rokitanského 62, 50003 Hradec Králové, Czech Republic)

  • Oskar A. Sultanov

    (Department of Differential Equations, Institute of Mathematics, Ufa Federal Research Center, Russian Academy of Sciences, Chernyshevsky str. 112, 450008 Ufa, Russia)

Abstract

We consider a singularly perturbed boundary value problem ( − ε 2 ∆ + ∇ V · ∇ ) u ε = 0 in Ω , u ε = f on ∂ Ω , f ∈ C ∞ ( ∂ Ω ) . The function V is supposed to be sufficiently smooth and to have the only minimum in the domain Ω . This minimum can degenerate. The potential V has no other stationary points in Ω and its normal derivative at the boundary is non-zero. Such a problem arises in studying Brownian motion governed by overdamped Langevin dynamics in the presence of a single attracting point. It describes the distribution of the points at the boundary ∂ Ω , at which the trajectories of the Brownian particle hit the boundary for the first time. Our main result is a complete asymptotic expansion for u ε as ε → + 0 . This asymptotic is a sum of a term K ε Ψ ε and a boundary layer, where Ψ ε is the eigenfunction associated with the lowest eigenvalue of the considered problem and K ε is some constant. We provide complete asymptotic expansions for both K ε and Ψ ε ; the boundary layer is also an infinite asymptotic series power in ε . The error term in the asymptotics for u ε is estimated in various norms.

Suggested Citation

  • Denis I. Borisov & Oskar A. Sultanov, 2020. "Complete Asymptotics for Solution of Singularly Perturbed Dynamical Systems with Single Well Potential," Mathematics, MDPI, vol. 8(6), pages 1-17, June.
  • Handle: RePEc:gam:jmathe:v:8:y:2020:i:6:p:949-:d:369530
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    References listed on IDEAS

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    1. Пигнастый, Олег & Koжевников, Георгий, 2019. "Распределенная Динамическая Pde-Модель Программного Управления Загрузкой Технологического Оборудования Производственной Линии [Distributed dynamic PDE-model of a program control by utilization of t," MPRA Paper 93278, University Library of Munich, Germany, revised 02 Feb 2019.
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