Solution of contrast structure type for a reaction-diffusion equation with discontinuous reactive term

Xiao Wu; Mingkang Ni

doi:10.3934/dcdss.2020341

Article Contents

2021, Volume 14, Issue 9: 3249-3266. Doi: 10.3934/dcdss.2020341

This issue Previous Article Higher order convergence for a class of set differential equations with initial conditions Next Article Chaotic oscillations of linear hyperbolic PDE with variable coefficients and implicit boundary conditions

Solution of contrast structure type for a reaction-diffusion equation with discontinuous reactive term

Xiao Wu^1, and
Mingkang Ni^2, ,

1.
School of Mathematical Sciences, East China Normal University, Shanghai 200000, P. R. China
2.
School of Mathematical Sciences, East China Normal University, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, Shanghai 200000, P. R. China

^* Corresponding author: Mingkang Ni
^* Corresponding author: Mingkang Ni

Received: May 31, 2019

Revised: September 30, 2019

Early access: April 2020

Published: September 2021

The second author is supported by the National Natural Science Foundation of China (No. 11871217) and the Science and Technology Commission of Shanghai Municipality (No. 18dz2271000)

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Abstract

In this paper, we consider the Dirichlet boundary value problem for a singularly perturbed reaction-diffusion equation with discontinuous reactive term. We use the asymptotic analysis to construct the formal asymptotic approximation of the solution with internal and boundary layers. The internal layer is located in the vicinity of a curve of the discontinuous reactive term. By using sufficiently precise lower and upper solutions, we prove the existence of a periodic solution and estimate the accuracy of its asymptotic approximation.

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Mathematics Subject Classification: Primary: 34D15, 34E15; Secondary: 34K13.

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Figure 1. the picture for the zero approximation $ U_{0}(x,t,\epsilon) $ for the solution $ u(x,t,\epsilon) $ of (50)

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