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Abstract <p>In this paper, the Dirichlet periodic boundary value problem of a class of singularly perturbed reaction-diffusion equations with different degree roots for discontinuous nonlinear reaction terms is studied. By introducing the nonstandard boundary layer function method, the formal asymptotic solutions of the inner layer and boundary layer with fractional order form are constructed when the degenerate equation has multiple roots. The existence of periodic solutions is proved by using sufficiently accurate upper and lower solutions, and the asymptotic approximation accuracy is estimated. The asymptotic stability of formal asymptotic solutions in the sense of Lyapunov is further proved. Finally, a numerical example is given to demonstrate our results.</p>

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The Existence and Stability of Periodic Solutions for Right-End Discontinuous Singularly Perturbed Reaction-Diffusion Problems with Multiple Roots of Degenerate Equations

  • 期刊论文

  • 发表日期：

    2025年12月1日

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  • Xin Shuai,
  • Mingkang Ni

摘要

Abstract

In this paper, the Dirichlet periodic boundary value problem of a class of singularly perturbed reaction-diffusion equations with different degree roots for discontinuous nonlinear reaction terms is studied. By introducing the nonstandard boundary layer function method, the formal asymptotic solutions of the inner layer and boundary layer with fractional order form are constructed when the degenerate equation has multiple roots. The existence of periodic solutions is proved by using sufficiently accurate upper and lower solutions, and the asymptotic approximation accuracy is estimated. The asymptotic stability of formal asymptotic solutions in the sense of Lyapunov is further proved. Finally, a numerical example is given to demonstrate our results.