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



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)


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.


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

2010 Mathematics Subject Classification

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

H. Jin was supported by the Guangdong Basic and Applied Basic Research Foundation (No. 2022B1515020032), Guangzhou Science and Technology Program (No. 202002030363), NSF of China (No. 11871226), and the Fundamental Research Funds for the Central Universities.

T. Xiang was funded by the NSF of China (No. 12071476 and 11871226) and the Research Funds of Renmin University of China (No. 2018030199).

Received 10 January 2022

Received revised 11 September 2022

Accepted 11 September 2022

Published 24 March 2023