Global asymptotic stability in a chemotaxis-growth model for tumor invasion

Kentarou Fujie

doi:10.3934/dcdss.2020011

Article Contents

2020, Volume 13, Issue 2: 203-209. Doi: 10.3934/dcdss.2020011

This issue Previous Article On a chemotaxis model with competitive terms arising in angiogenesis Next Article Boundedness in a quasilinear fully parabolic Keller-Segel system via maximal Sobolev regularity

Global asymptotic stability in a chemotaxis-growth model for tumor invasion

Kentarou Fujie^,

Department of Mathematics, Tokyo University of Science, Tokyo 162-8601, Japan

* Corresponding author: Kentarou Fujie
* Corresponding author: Kentarou Fujie

Received: April 30, 2017

Revised: November 30, 2017

Published: February 2020

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Abstract

This paper presents global existence and asymptotic behavior of solutions to the chemotaxis-growth system

$ \left\{ \begin{array}{l} u_t = \Delta u - \nabla \cdot (u\nabla v) + ru -\mu u^\alpha, \qquad x\in \Omega, \ t>0, \\ \ v_t = \Delta v + wz, \qquad x\in \Omega, \ t>0, \\ \ w_t = -wz, \qquad x\in \Omega, \ t>0, \\ \ z_t = \Delta z - z + u, \qquad x\in \Omega, \ t>0, \end{array} \right. $

in a smoothly bounded domain $ \Omega \subset \mathbb{R}^n $, $ n \le 3 $, where $ r>0 $, $ \mu>0 $ and $ \alpha>1 $. Without the logistic source $ ru-\mu u^\alpha $, the stabilization of this system has been shown by Fujie, Ito, Winkler and Yokota (2016), whereas especially about asymptotic behavior, the logistic source disturbs applying this method directly. In the present paper, a way out of this difficulty is introduced and the asymptotic behavior of solutions to the system with logistic source is precisely determined.

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Mathematics Subject Classification: Primary: 35B40, 35Q92; Secondary: 92C17.

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