Global attractor for a partly dissipative reaction-diffusion system with discontinuous nonlinearity

Jia-Cheng Zhao; Zhong-Xin Ma

doi:10.3934/dcdsb.2022103

Article Contents

2023, Volume 28, Issue 2: 893-908. Doi: 10.3934/dcdsb.2022103

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Global attractor for a partly dissipative reaction-diffusion system with discontinuous nonlinearity

Jia-Cheng Zhao^1, and
Zhong-Xin Ma^1,2, ,

1.
Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
2.
Department of Mathematics, Zhejiang University of Science and Technology, Hangzhou 310023, China

^*Corresponding author: Zhong-Xin Ma
^*Corresponding author: Zhong-Xin Ma

Received: December 31, 2021

Revised: March 31, 2022

Early access: May 2022

Published: February 2023

Both of the authors are supported by the NSFC of China (No. 11971317)

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Abstract

In this paper, we consider a partly dissipative reaction-diffusion system with discontinuous nonlinearity in the form

$ \begin{equation*} \left\{\begin{array}{ll} u_t-\Delta u+u+w\in H_0(u-a), \\ w_t-\epsilon(u-\gamma w) = 0, \end{array}\right. \end{equation*} $

where $ H_0 $ is a multi-valued function of Heaviside type. This type of system is used for describing the generation and transmission of electrical signals in neuroscience. We first present an existence result on global solutions. Then, we prove that the system possesses a global attractor having the $ H^r\times H^r $-regularity $ (0\leq r<2) $. Moreover, by showing the Kneser property for the system, the global attractor is proved to be connected. The main characteristic of the system is that the linear part cannot be represented as the subdifferential of a compact-type function.

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Mathematics Subject Classification: Primary: 35B41, 35R70; Secondary: 35K57, 35B40.

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