Global bounded solution to a chemotaxis-convection model of capillary-sprout growth during tumor angiogenesis
type: Item
instance of: scholarly article
title: Global bounded solution to a chemotaxis-convection model of capillary-sprout growth during tumor angiogenesis
zbMATH Open document ID: 1470.35184
author: Yanyan Li
published in: Journal of Mathematical Analysis and Applications
publication date: 2021-02-25T00:00:00Z
review text: The authors study the chemotaxis-convection system \[\begin{cases} u_t = d_1 \Delta u - \nabla \cdot (u \nabla v) + \nabla \cdot (u \nabla w) + f(u), & \; (x,t) \in \Omega \times (0,\infty), \\ v_t = d_2 \Delta v + \nabla \cdot (v \nabla w) + ku - \mu v, & \; (x,t) \in \Omega \times (0,\infty), \\ w_t = d_3 \Delta w + ru - \delta w, & \; (x,t) \in \Omega \times (0,\infty), \end{cases} \] endowed with homogeneous Neumann boundary conditions and nonnegative initial data \((u_0,v_0,w_0)\) with \(u_0 \in W^{1,\infty}(\Omega)\), \(v_0, w_0 \in W^{2,\infty}(\Omega)\) and \(u_0 \not\equiv 0\), where \(\Omega \subset \mathbb{R}^2\) is a bounded domain with smooth boundary. Moreover, \(d_i\), \(k\), \(r\), \(\mu\), \(\delta\) are positive constants and \(f \in C^1([0,\infty))\) satisfies \(f(0) \ge 0\) and \(f(s) \le a - b s^m\) for \(s>0\) with some \(a\ge 0\), \(b>0\), and \(m>3\). The system describes the branching of capillary sprouts during angiogenesis, where \(u\), \(v\), and \(w\) denote the densities of endothelial cells, adhesive sites, and matrix, respectively. Requiring the assumptions given above (in particular \(m>3\)), the authors prove the existence of a unique global and bounded classical solution to the above system such that \(u,v,w \in L^\infty((0,\infty); W^{1,4}(\Omega))\). In the proof the authors mainly establish several a priori estimates which are combined in a clever way to result in the uniform boundedness mentioned above. In particular, results on maximal Sobolev regularity of the Neumann heat semigroup are used to prove some of these estimates.
reviewed by: Christian Stinner
Mathematics Subject Classification ID: 35K51
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35Q92
Mathematics Subject Classification ID: 92C17
zbMATH DE Number: 7315349
zbMATH Keywords: chemotaxis-convection model
zbMATH Keywords: growth source
zbMATH Keywords: classical solution
zbMATH Keywords: global boundedness
zbMATH Keywords: maximal Sobolev regularity
zbMATH Keywords: Neumann heat semigroup
MaRDI profile type: Publication
full work available at URL: j.jmaa.2020.124665
OpenAlex ID: W3092035732
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DOI: 10.1016/J.JMAA.2020.124665
description: scientific article; zbMATH DE number 7315349