Existence and continuity of uniform exponential attractors of the singularity perturbed Hodgkin-Huxley system
type: Item
instance of: scholarly article
title: Existence and continuity of uniform exponential attractors of the singularity perturbed Hodgkin-Huxley system
zbMATH Open document ID: 0912.35030
published in: Journal of Differential Equations
publication date: 1999-02-03T00:00:00Z
review text: It is not possible in this short review to discuss satisfactorily the authors paper which comprises seventy pages. We thus have to content us with a few indications. The author studies rate of attraction and fractal dimension of two systems, both intimately connected with the Hodgkin-Huxley system. The first of these is \[ \left. \begin{aligned} u_t &-\Delta u+ f_1(w)u= f_2(w))\\ w_t &+ h_1(u)w= h_2(u)\end{aligned} \right\}\quad\text{in }\Omega,\tag{1} \] \[ \partial_nu= 0\quad \text{on }\partial\Omega, \] where \(\Omega\subseteq \mathbb{R}^N\) is a smooth bounded domain and \(\partial_n u(P)\) the derivative with respect to the outer unit normal at \(P\in\partial\Omega\). The second system reads as follows: \[ \left. \begin{aligned} \varepsilon u_{tt} &+ (\varepsilon f_3(w)+ 1)u_t- \Delta u+ f_1(w)u= f_3(w)\\ w_t &+ h_1(u)w= h_2(u)\end{aligned} \right\}\quad\text{in }\Omega,\tag{2} \] \[ \partial_nu= 0\quad\text{on }\partial\Omega. \] The functions \(f_i\), \(h_j\) together with their derivatives are assumed to be continuous and uniformly bounded and such that (1) admits a positive invariant region. Systems (1), (2) give rise to semiflows, namely \[ \begin{aligned} S_\varepsilon(t)(u_0, v_0, w_0) &= (u^\varepsilon(t), u^\varepsilon_t(t), w^\varepsilon(t)),\\ S(t)(u_0, w_0) & = (u(t), w(t)).\end{aligned} \] Here \(u^\varepsilon(t)\), \(u^\varepsilon_t(t)\), \(w^\varepsilon(t)\) is the solution to (2) for initial conditions \(u_0\), \(u_t(0)= v_0\), \(w_0\); likewise with \(S(t)\). Both \(S(t)\) and \(S_\varepsilon(t)\) act on suitable function spaces. In order to express the main result one has to replace \(S_\varepsilon(t)\) by an extension \(S^\varepsilon(t)\) which acts on four-vectors: \[ S^\varepsilon(t)(u_0, v_0,\sqrt\varepsilon v_0,w_0)= (u^\varepsilon(t), u^\varepsilon_t(t), \sqrt\varepsilon u^\varepsilon_t(t), w^\varepsilon(t)). \] The principal result now states that for \(\varepsilon\) sufficiently small, \(^\varepsilon(t)\), \(t\geq 0\) has an exponentia…
reviewed by: Bruno Scarpellini
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 37C70
Mathematics Subject Classification ID: 35B25
zbMATH DE Number: 1206647
zbMATH Keywords: rate of attraction
zbMATH Keywords: fractal dimension
author: C. Galusinski
MaRDI profile type: Publication
full work available at URL: jdeq.1997.3357
OpenAlex ID: W2093443583
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DOI: 10.1006/JDEQ.1997.3357
description: scientific article; zbMATH DE number 1206647