Global boundedness in a quasilinear two-species attraction-repulsion chemotaxis system with two chemicals
type: Item
instance of: scholarly article
title: Global boundedness in a quasilinear two-species attraction-repulsion chemotaxis system with two chemicals
author: Miaoqing Tian
author: Shujuan Wang
author: Xia Xiao
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2022-10-10T00:00:00Z
review text: The authors study the quasilinear attraction-repulsion chemotaxis system with two species and two chemicals \[ \begin{cases} u_t = \nabla \cdot (D_1(u) \nabla u) - \nabla \cdot(\Phi_1(u) \nabla v), & \quad (x,t) \in \Omega \times (0,\infty),\\ 0 = \Delta v - v + w^{\gamma_1}, & \quad (x,t) \in \Omega \times (0,\infty), \\ w_t = \nabla \cdot (D_2(w) \nabla w) + \nabla \cdot(\Phi_2(w) \nabla z), & \quad (x,t) \in \Omega \times (0,\infty),\\ 0 = \Delta z - z + u^{\gamma_2}, & \quad (x,t) \in \Omega \times (0,\infty), \end{cases} \] endowed with homogeneous Neumann boundary conditions and nonnegative initial data \(u_0, w_0 \in C(\overline{\Omega})\). Moreover, it is assumed that \(\Omega \subset \mathbb{R}^N\) is a bounded domain with smooth boundary with \(N \ge 2\) and that \(D_i, \Phi_i \in C^2([0,\infty))\) for \(i=1,2\) satisfy \(D_i(s) \ge (s+1)^{p_i}\) and \(\Phi_i(s) \ge 0\) for all \(s \ge 0\) as well as \(\Phi_i (s) \le \chi_i s^{q_i}\) for all \(s \ge s_0\) with some \(\gamma_i, \chi_i >0\), \(p_i, q_i \in \mathbb{R}\), and \(s_0 >1\). Here \(u\) and \(w\) denote the densities of two populations and \(v\) and \(z\) the concentration of two chemicals. While \(u\) is attracted by \(v\), which is produced by \(w\), \(w\) is repelled by \(z\), which is produced by \(u\). The authors prove the existence of a unique global classical solution to the above system provided that one of the following assumptions is fulfilled: (\(\gamma_1 < \frac{2}{N}\) and \(q_1 + \gamma_1 - p_1 < 1 + \frac{2}{N}\)) or (\(\gamma_2 < \frac{4}{N}\), \(\gamma_2 \le 1\), and \(q_2 + \gamma_2 - p_2 < 1 + \frac{4}{N}\)) or (\(q_i + \gamma_i - p_j \le 1+ \frac{2}{N}\) for \(i,j=1,2\)). The proof mainly relies on appropriate a priori estimates for \(u\) and \(w\).
reviewed by: Christian Stinner
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K51
Mathematics Subject Classification ID: 35K59
Mathematics Subject Classification ID: 92C17
zbMATH DE Number: 7599011
zbMATH Keywords: self-diffusion dominance
zbMATH Keywords: quasilinear attraction-repulsion chemotaxis
MaRDI profile type: Publication
full work available at URL: dcdsb.2022071
OpenAlex ID: W4225656852
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DOI: 10.3934/DCDSB.2022071
description: scientific article; zbMATH DE number 7599011