华东师范大学学报（自然科学版） >

2022 , Vol. 2022 >Issue 1: 1 - 9

DOI: https://doi.org/10.3969/j.issn.1000-5641.2022.01.001

数学

一类右端不连续的奇摄动二阶半线性微分方程解的稳定性态

  • 刘乐思 ,
  • 倪明康
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  • 1. 华东师范大学 数学科学学院, 上海　200241

收稿日期: 2021-01-18

  网络出版日期: 2022-01-18

基金资助

the National Nature Science Foundation of China (No.11871217) and the Science and Technology Commission of Shanghai Municipality (No.18dz2271000)

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Stability of the solution to a singularly perturbed semilinear second-order differential equation with discontinuous right-hand side

  • Aleksei LIUBAVIN ,
  • Mingkang NI
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  • 1. School of Mathematical Sciences, East China Normal University, Shanghai　200241, China

Received date: 2021-01-18

  Online published: 2022-01-18

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摘要

本文研究了一类右端不连续的反应扩散方程的稳态问题. 基于空间对照结构理论, 通过求解Sturm-Liouville问题构造了特征值和特征函数的渐近展开式, 并给出了该表达式的余项估计以及稳态解稳定的充分性条件.

关键词： 奇摄动; 渐近逼近; 稳定性; 反应扩散方程

本文引用格式

刘乐思 , 倪明康 . 一类右端不连续的奇摄动二阶半线性微分方程解的稳定性态[J]. 华东师范大学学报（自然科学版）, 2022 , 2022(1) : 1 -9 . DOI: 10.3969/j.issn.1000-5641.2022.01.001

Abstract

In this paper, a stationary problem for the reaction-diffusion equation with a discontinuous right-hand side is considered. Based on ideas from contrast structure theory, the asymptotic representations for eigenvalues and eigenfunctions are constructed by solving a Sturm-Liouville problem and an estimation of the remainder is obtained. Moreover, a sufficient condition which guarantees the stability of the solution to this task is established.

Key words： singular perturbations; asymptotic approximations; stability; reaction-diffusion equation

参考文献

1 BUTUZOV V F, VASILEVA A B. Asymptotic of a solution of contrast-structure type [J]. Mathematical Notes of the Academy of Sciences of the USSR, 1987, 42: 956-961.
2 VASILEVA A B. On the stability of the contrast structures. Mathematical Modeling, 1991, (3/4): 114- 123.
3 VASILEVA A B. On the question of the stability of solutions to the type of contrast structures. Mathematical Modeling, 1990, (1/2): 119- 125.
4 NEFEDOV N N, NI M K. Internal layers in the one-dimensional reaction-diffusion equation with a discontinuous reactive term. Computational Mathematics & Mathematical Physics, 2015, 55 (12): 2001- 2007.
5 HENRY D. Geometric Theory of Semilinear Parabolic Equations [M]. Berlin: Springer-Verlag, 1981: 350.
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