Weak solutions to a class of signal-dependent motility Keller-Segel systems with superlinear damping
type: Item
instance of: scholarly article
title: Weak solutions to a class of signal-dependent motility Keller-Segel systems with superlinear damping
zbMATH Open document ID: 1483.35109
author: Wenbin Lyu
published in: Journal of Differential Equations
publication date: 2022-01-20T00:00:00Z
review text: The existence of global non-negative weak solutions to the Keller-Segel chemotaxis system with logistic term \begin{align*} \partial_t u & = \mathrm{div}(D(v) \nabla u - u S(v)\nabla v) + f(u) \text{ in } (0,\infty)\times\Omega, \\ \partial_t v & = \Delta v - v + u\text{ in } (0,\infty)\times\Omega, \end{align*} supplemented with homogeneous Neumann boundary conditions and initial conditions \(u_0\in C(\bar{\Omega})\), \(u_0\ge 0\), \(u_0\not\equiv 0\), and \(v_0\in W^{2,\infty}(\Omega)\), \(v_0>0\), is investigated. Here, \(\Omega\) is a bounded domain of \(\mathbb{R}^n\), \(n\ge 1\), with smooth boundary, the functions \(D\) and \(S\) belong to \(C^3([0,\infty))\) and satisfy \[ 0 < k \le D(s) \le K\,, \quad |S(s)|\le K\,, \qquad s\ge x_0\,, \] for some \(x_0>0\), and the logistic term \(f\in C^1([0,\infty))\) is such that \(f(0)=0\) and \(f(s) s^{-l} \to -\infty\) as \(s\to\infty\) for some \(l>\max\left\{ \frac{2(n+2)}{n+4} , \frac{2(n-1)}{n} \right\}\). The proof is performed by a compactness approach, the key tool being the use of the superlinear dissipation induced by the logistic term to derive suitable estimates on \(v\) and then on \(u\).
Mathematics Subject Classification ID: 35K51
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35Q92
Mathematics Subject Classification ID: 35K59
Mathematics Subject Classification ID: 35D30
zbMATH DE Number: 7460670
zbMATH Keywords: chemotaxis
zbMATH Keywords: global existence
zbMATH Keywords: logistic term
zbMATH Keywords: weak solution
reviewed by: Philippe Laurençot
MaRDI profile type: Publication
full work available at URL: j.jde.2021.12.020
OpenAlex ID: W4206338911
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DOI: 10.1016/J.JDE.2021.12.020
description: scientific article; zbMATH DE number 7460670