Stability of solutions of chemotaxis equations in reinforced random walks
type: Item
instance of: scholarly article
title: Stability of solutions of chemotaxis equations in reinforced random walks
zbMATH Open document ID: 1025.35005
DOI: 10.1016/S0022-247X(02)00147-6
author: Yanyan Li
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-11-28T00:00:00Z
review text: The authors consider the following nonlinear system of differential equations consisting of one parabolic equation and one ordinary differential equation: \[ \begin{aligned} \frac{\partial p}{\partial t}&= \operatorname {div}(\nabla p-p\chi (w) \nabla w), \quad x\in\Omega, \;t\geq 0,\\ \frac{\partial w}{\partial t}&= g(p, w), \quad x\in\Omega, \;t\geq 0,\end{aligned} \] with some boundary condition. Here \(\Omega\) is a domain in a Euclidean space, \(g\) and \(\chi\) are given scalar functions. This system arises in chemotaxis a process whereby living organisms respond to chemical substance by moving toward higher, or lower, concentrations of the chemical substance, or by aggregating or dispersing. It is proved that stationary solutions of the system are asymptotically stable.
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 92C17
Mathematics Subject Classification ID: 60K40
Mathematics Subject Classification ID: 35K60
zbMATH DE Number: 1836782
zbMATH Keywords: parabolic equation
zbMATH Keywords: ordinary differential equation
zbMATH Keywords: system of equations
zbMATH Keywords: stationary solutions
reviewed by: Michael I. Gil'
MaRDI profile type: Publication
cites work: Q3890476
cites work: Biased random walk models for chemotaxis and related diffusion approximations
cites work: Continuous and discrete mathematical models of tumor-induced angiogenesis
cites work: Reinforced random walk
cites work: Mathematical Analysis of a Model for the Initiation of Angiogenesis
cites work: Chemotactic collapse for the Keller-Segel model
cites work: Singularity patterns in a chemotaxis model
cites work: Q4398321
cites work: On Explosions of Solutions to a System of Partial Differential Equations Modelling Chemotaxis
cites work: Model for chemotaxis
cites work: Q5562267
cites work: Erratum: ``Mathematical modeling of capillary formation and development in tumor angiogenesis: penetration into the stroma''
cites work: A System of Reaction Diffusion Equations Arising in the Theory of Reinforced Random Walks
cites work: A mathematical model for the roles of pericytes and macrophages in the initiation of angiogenesis. I: The role of protease inhibitors in preventing angiogenesis
cites work: Q4354424
cites work: Q4872331
cites work: Aggregation, Blowup, and Collapse: The ABC's of Taxis in Reinforced Random Walks
cites work: Stationary Solutions of Chemotaxis Systems
cites work: The Derivation of Chemotaxis Equations as Limit Dynamics of Moderately Interacting Stochastic Many-Particle Systems
cites work: A critical parabolic estimate and application to nonlocal equations arising in chemotaxis
description: scientific article; zbMATH DE number 1836782