Dynamics of Partial Differential Equations

Volume 18 (2021)

Number 1

Global dynamics of partly diffusive Hindmarsh–Rose equations in neurodynamics

Pages: 33 – 47

DOI: https://dx.doi.org/10.4310/DPDE.2021.v18.n1.a3

Authors

Chi Phan (Department of Mathematics and Statistics, Sam Houston State University, Huntsville, Texas, U.S.A.)

Yuncheng You (Department of Mathematics and Statistics, University of South Florida, Tampa, Fl., U.S.A.)

Jianzhong Su (Department of Mathematics, University of Texas, Arlington, Tx., U.S.A.)

Abstract

Global dynamics of the partly diffusive Hindmarsh–Rose equations as a new mathematical model in neurodynamics is presented and studied in this paper. The existence of global attractor for the solution semiflow is proved through uniform estimates showing the higher-order dissipative property and the ultimate compactness by the new approach of Kolmogorov–Riesz theorem.

Keywords

diffusive Hindmarsh–Rose equations, neurodynamics, global attractor, absorbing property, ultimate compactness, Kolmogorov–Riesz theorem

2010 Mathematics Subject Classification

35B41, 35K58, 35Q92, 37N25, 92C20

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Jianzhong Su is partially supported by USDA Grant number 2018-38422-28564.

Received 15 June 2020

Published 19 February 2021