Global attractor for a chemotaxis model with prevention of overcrowding
type: Item
instance of: scholarly article
title: Global attractor for a chemotaxis model with prevention of overcrowding
zbMATH Open document ID: 1065.35072
author: Dariusz Wrzosek
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2005-01-11T00:00:00Z
review text: The author studies the following system of two parabolic equations modelling the chemotaxis with prevention of overcrowding: \[ \begin{cases} \frac{\partial u}{\partial t}=\nabla.(d_u(q(u)-q'(u)u)\nabla u-uq(u)\chi(v)\nabla v)+uf(u),\\ \frac{\partial v}{\partial t}=d_v\Delta v-g_1(u)-vg_2(v),\end{cases} \tag{1} \] which is considered in a bounded domain of \(\mathbb R^n\) equipped by Robin boundary conditions. Here \(d_u\) and \(d_v\) are positive diffusion coefficients, and \(q(u)\), \(g_1(u)\), \(f(u)\), \(g_2(v)\) and \(\chi(v)\) are assumed to satisfy the assumptions which gives the nondegenerate parabolicity of the problem considered and guarantee that the domain \(V_M:=\{v\geq0,\;\;0\leq u\leq U_M\}\) (for some fixed positive \(U_M\)) is invariant with respect to the flow generated by (1). To be more precise, the last assumption requires: \(g_1(u)\geq0,\;\;f(U_M)\leq 0,\;\;q(U_M)=0\) and, for the nondegeneracy, \(d(\xi):=d_u(q(\xi)-q'(\xi)\xi)>0,\;\;\xi\in[0,U_M].\) Then, under some more technical assumptions on the nonlinearities, the author verifies the existence of a global attractor \(\mathcal A\) for equation (1) in the domain \(V_M\) (equipped with the appropriate topology).
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 37L30
Mathematics Subject Classification ID: 35K50
zbMATH DE Number: 2126038
zbMATH Keywords: volume filling model
zbMATH Keywords: quasilinear parabolic system
reviewed by: S. V. Zelik
MaRDI profile type: Publication
full work available at URL: j.na.2004.08.015
OpenAlex ID: W2049223241
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DOI: 10.1016/J.NA.2004.08.015
description: scientific article; zbMATH DE number 2126038