On a nonlocal aggregation model with nonlinear diffusion
type: Item
instance of: scholarly article
title: On a nonlocal aggregation model with nonlinear diffusion
zbMATH Open document ID: 1209.35070
author: Dong Li
author: Xiaoyi Zhang
published in: Discrete and Continuous Dynamical Systems
publication date: 2010-05-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/0902.2017
review text: This paper concerns a nonlocal aggregation equation with nonlinear degenerate diffusion which arises from the study of biological aggregation dynamics: \[ \partial_t u + \partial_x(uv) = 0\;\text{ for } (t,x) \in (0,\infty) \times \mathbb R, \quad\text{and}\quad u(0,x) = u_0(x)\;\text{ for } x \in \mathbb R, \] where \(v = \partial_x(K * u) - ru\partial_xu\), \(K= e^{-|x|}\), \(r\) is a positive constant, and \(u_0 \in \bigcap_{m=0}^\infty H^m(\mathbb R)\). The solution \(u = u(t,x)\) represents population density in biology or particle density in material science. This equation is derived by \textit{Bertozzi, Lewis} and \textit{Topaz} [Bull. Math. Bio. 68, 1601--1623, (2006)]. By using the \(L^\infty\) energy method introduced by \textit{M. Ôtani} and \textit{Y. Sugiyama} [J. Math. Soc. Japan 53, No.~4, 745--789 (2001; Zbl 1024.35056)], the authors show the existence and uniqueness of smooth local solutions for initial data which is not necessarily nonnegative. It is also proved that any nonzero smooth initial data with compact support will lead to blowup of the gradient in finite time.
reviewed by: Shigeru Sakaguchi
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35K59
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 35Q92
Mathematics Subject Classification ID: 35R09
Mathematics Subject Classification ID: 35B44
zbMATH DE Number: 5704404
zbMATH Keywords: degenerate diffusion
zbMATH Keywords: energy method
zbMATH Keywords: local solution
zbMATH Keywords: blowup of gradient
MaRDI profile type: Publication
OpenAlex ID: W2033108664
arXiv ID: 0902.2017
DOI: 10.3934/DCDS.2010.27.301
Recommended article: Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion
Recommended article: Q5270231
Recommended article: Q2989871
Recommended article: Stationary States of an Aggregation Equation with Degenerate Diffusion and Bounded Attractive Potential
Recommended article: Uniqueness and Nonuniqueness of Steady States of <scp>Aggregation‐Diffusion</scp> Equations
Recommended article: Existence and uniqueness of a weak solution of a nonlocal aggregation equation with degenerate diffusion of general form
Recommended article: Global minimizers for free energies of subcritical aggregation equations with degenerate diffusion
Recommended article: Stationary states of quadratic diffusion equations with long-range attraction
Recommended article: Localized Cluster Solutions of Nonlinear Degenerate Diffusion Equations Arising in Population Dynamics
Recommended article: Nonlinear aggregation-diffusion equations: radial symmetry and long time asymptotics
description: scientific article