Abstract
We consider singularly perturbed reaction–diffusion equations with singularly perturbed Neumann boundary conditions. We establish the existence of a time-periodic solution u(x, t, ε) with boundary layers and derive conditions for their asymptotic stability. The boundary layer part of u(x, t, ε) is of order one, which distinguishes our case from the case of regularly perturbed Neumann boundary conditions, where the boundary layer is of order ε. Another peculiarity of our problem is that — in contrast to the case of Dirichlet boundary conditions — it may have several asymptotically stable time-periodic solutions, where these solutions differ only in the description of the boundary layers. Our approach is based on the construction of sufficiently precise lower and upper solutions.
Highlights
For a singularly perturbed parabolic problem with Dirichlet conditions we prove the existence of a solution periodic in time and with boundary layers at both ends of the space interval in the case that the degenerate equation has a double root
We also investigate the stability of this solution and the corresponding region of attraction
N., Recke L., Schneider K., "Asymptotics, Stability and Region of Attraction of a Periodic Solution to a Singularly Perturbed Parabolic Problem in Case of a Multiple Root of the Degenerate Equation", Modeling and Analysis of Information Systems, 23:3 (2016), 248–258
Summary
В данной работе задача (1.1) – (1.3) исследуется при условии, что вырожденное уравнение (1.4) имеет двукратный корень. Что аналогичная задача о погранслойном T -периодическом по времени решении уравнения (1.1) с функцией f (u, x, t, ε) вида (1.5) и краевыми условиями Неймана была рассмотрена в [3]. В этом случае, в отличие от рассматриваемого в данной работе, асимптотика строится с помощью стандартного алгоритма и пограничные слои имеют однозонный характер с экспоненциальным убыванием пограничных функций, как и в случае простого корня вырожденного уравнения. Рассматриваемая в данной работе краевая задача Дирихле развивает результаты работы авторов по построению асимптотики для начальной задачи [4] на более сложный класс, а также методы обоснования асимптотики, исследования устойчивости и области влияния устойчивого решения работ [5], [6], [7].
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