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

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.