Communications in Mathematical Sciences

Volume 21 (2023)

Number 4

Global dynamics in a chemotaxis model describing tumor angiogenesis with/without mitosis in any dimension

Pages: 1055 – 1095

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n4.a7

Authors

Jiawei Chu (School of Mathematics, South China University of Technology, Guangzhou, China)

Hai-Yang Jin (School of Mathematics, South China University of Technology, Guangzhou, China)

Tian Xiang (Institute for Mathematical Sciences and School of Mathematics, Renmin University of China, Beijing, China)

Abstract

In this work, we study the following Neumann-initial boundary value problem for a three-component chemotaxis model describing tumor angiogenesis:\[\left \{\begin{array}u_t = \Delta u-\chi \nabla \cdot (u \nabla v)+ \xi_1 \nabla \cdot (u \nabla w)+u(a-\mu u^\theta ), & x \in \Omega , t \gt 0, \\v_t = d \Delta v+ \xi_2 \nabla \cdot (v \nabla w)+u-v, & x \in \Omega , t \gt 0, \\0= \Delta w+u-\overline{u}, \int_{\Omega} w=0, \overline{u}:= \frac{1}{\lvert\Omega\vert} \int_{\Omega} u, & x \in \Omega , t \gt 0, \\\frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu} =0, & x \in \partial \Omega , t \gt 0, \\u(x, 0)=u_0(x), v(x, 0)=v_0(x), & x \in \Omega ,\end{array}\right .\]in a bounded smooth but not necessarily convex domain $\Omega \subset \mathbb{R}^n (n \geq 2)$ with model parameters $\chi_1, \chi_2, d, \theta \gt 0, a, \chi , \mu \geq 0$. Based on subtle energy estimates, we first identify two positive constants $\chi_0$ and $\mu_0$ such that the above problem allows only global classical solutions with qualitative bounds provided one of the following conditions holds:\[\textrm{(1)} \; \xi_1 \geq \xi_0 \chi^2 \; \mathrm{;}\quad \textrm{(2)} \; \theta = 1, \mu \geq \max {\biggl\lbrace 1, \chi^{\frac{8+2n}{5+n}} \biggr\rbrace} \mu_0 \chi^{\frac{2}{5+n}} \; \mathrm{;}\quad \textrm{(3)} \; \theta > 1, \mu > 0 \; \mathrm{.}\]Then, due to the obtained qualitative bounds, upon deriving higher order gradient estimates, we show exponential convergence of bounded solutions to the spatially homogeneous equilibrium (i) for $\mu$ large if $\mu \gt 0$, (ii) for $d$ large if $a=\mu=0$ and (iii) for merely $d \gt 0$ if $\chi=a=\mu =0$. As a direct consequence of our findings, all solutions to the above system with $\chi=a=\mu=0$ are globally bounded and they converge to constant equilibrium, and therefore, no patterns can arise.

Keywords

chemotaxis, tumor angiogenesis, convection, qualitative boundedness, global stability

2010 Mathematics Subject Classification

35A01, 35B40, 35K57, 35Q92, 92C17

Received 10 January 2022

Received revised 11 September 2022

Accepted 11 September 2022

Published 24 March 2023