数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (3): 1046-1063.doi: 10.1007/s10473-024-0316-7

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DYNAMICS FOR A CHEMOTAXIS MODEL WITH GENERAL LOGISTIC DAMPING AND SIGNAL DEPENDENT MOTILITY

Xinyu Tu1,2, Chunlai Mu3,*, Shuyan Qiu4, Jing Zhang5   

  1. 1. School of Mathematics and Statistics, Southwest University, Chongqing 400715, China;
    2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong, China;
    3. College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China;
    4. School of Sciences, Southwest Petroleum University, Chengdu 610500, China;
    5. College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China
  • 收稿日期:2023-01-21 修回日期:2023-06-02 出版日期:2024-06-25 发布日期:2024-05-21

DYNAMICS FOR A CHEMOTAXIS MODEL WITH GENERAL LOGISTIC DAMPING AND SIGNAL DEPENDENT MOTILITY

Xinyu Tu1,2, Chunlai Mu3,*, Shuyan Qiu4, Jing Zhang5   

  1. 1. School of Mathematics and Statistics, Southwest University, Chongqing 400715, China;
    2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong, China;
    3. College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China;
    4. School of Sciences, Southwest Petroleum University, Chengdu 610500, China;
    5. College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China
  • Received:2023-01-21 Revised:2023-06-02 Online:2024-06-25 Published:2024-05-21
  • Contact: *Chunlai Mu, E-mail:clmu2005@163.com
  • About author:Xinyu Tu,E-mail:xinyutututu@163.com;Shuyan Qiu, E-mail:shuyanqiu0701@126.com; Jing Zhang, E-mail:zj188838@163.com
  • Supported by:
    Tu's work was supported by the NSFC (12301260), the Hong Kong Scholars Program (XJ2023002, 2023-078), the Double First-Class Construction-Talent Introduction of Southwest University (SWU-KR22037) and the Chongqing Post-Doctoral Fund for Staying in Chongqing (2022); Mu's work was partially supported by the NSFC (12271064, 11971082), the Chongqing Talent Support Program (cstc2022ycjh-bgzxm0169), the Natural Science Foundation of Chongqing (cstc2021jcyj-msxmX1051), the Fundamental Research Funds for the Central Universities (2020CDJQY-Z001, 2019CDJCYJ001) and the Key Laboratory of Nonlinear Analysis and its Applications (Chongqing University), Ministry of Education, and Chongqing Key Laboratory of Analytic Mathematics and Applications; Qiu's work was supported by the NSFC (12301261), the Scientific Research Starting Project of SWPU (2021QHZ016), the Sichuan Science and Technology Program (2023NSFSC1365) and the Nanchong Municipal Government-Universities Scientific Cooperation Project(SXHZ045); Zhang's work was supported by the China Scholarship Council (202206050060) and the Graduate Research and Innovation Foundation of Chongqing (CYB22044).

摘要: In this paper, we consider the fully parabolic chemotaxis system with the general logistic source
$\begin{eqnarray*}\left\{\begin{array}{llll}u_t= \Delta(\gamma(v) u )+\lambda u-\mu u^{\kappa},~~~ &x \in \Omega, ~t>0,\\ v_t= \Delta v+wz, &x \in \Omega, ~t>0,\\w_t= -wz, &x \in \Omega, ~t>0,\\z_t= \Delta z - z+ u, &x\in \Omega, ~t>0,\\\end{array}\right.\end{eqnarray*}$
where $\Omega\subset \mathbb{R}^n (n\geq 1)$ is a smooth and bounded domain, $\lambda\geq 0, \mu\geq 0, \kappa>1$, and the motility function satisfies that $\gamma(v)\in C^3([0, \infty))$, $\gamma(v)>0$, $\gamma{'}(v)\leq0$ for all $v\geq 0$. Considering the Neumann boundary condition, we obtain the global boundedness of solutions if one of the following conditions holds: (i) $ \lambda=\mu=0, 1\leq n\leq 3; $(ii) $ \lambda> 0, \mu>0, ~\text{combined with}~ \kappa>1, 1\leq n\leq 3 ~~\text{or}~~\kappa>\frac{n+2}{4}, n>3. $ Moreover,we prove that the solution $(u, v, w, z)$ exponentially converges to the constant steady state $\left(\left(\frac{\lambda}{\mu}\right)^{\frac{1}{\kappa-1}}, \frac{\int_{\Omega}v_0{\rm d}x+\int_{\Omega}w_0{\rm d}x}{|\Omega|}, 0, \left(\frac{\lambda}{\mu}\right)^{\frac{1}{\kappa-1}}\right)$.

关键词: chemotaxis, signal-dependent motility, logistic source, boundedness, asymptotic behavior

Abstract: In this paper, we consider the fully parabolic chemotaxis system with the general logistic source
$\begin{eqnarray*}\left\{\begin{array}{llll}u_t= \Delta(\gamma(v) u )+\lambda u-\mu u^{\kappa},~~~ &x \in \Omega, ~t>0,\\ v_t= \Delta v+wz, &x \in \Omega, ~t>0,\\w_t= -wz, &x \in \Omega, ~t>0,\\z_t= \Delta z - z+ u, &x\in \Omega, ~t>0,\\\end{array}\right.\end{eqnarray*}$
where $\Omega\subset \mathbb{R}^n (n\geq 1)$ is a smooth and bounded domain, $\lambda\geq 0, \mu\geq 0, \kappa>1$, and the motility function satisfies that $\gamma(v)\in C^3([0, \infty))$, $\gamma(v)>0$, $\gamma{'}(v)\leq0$ for all $v\geq 0$. Considering the Neumann boundary condition, we obtain the global boundedness of solutions if one of the following conditions holds: (i) $ \lambda=\mu=0, 1\leq n\leq 3; $(ii) $ \lambda> 0, \mu>0, ~\text{combined with}~ \kappa>1, 1\leq n\leq 3 ~~\text{or}~~\kappa>\frac{n+2}{4}, n>3. $ Moreover,we prove that the solution $(u, v, w, z)$ exponentially converges to the constant steady state $\left(\left(\frac{\lambda}{\mu}\right)^{\frac{1}{\kappa-1}}, \frac{\int_{\Omega}v_0{\rm d}x+\int_{\Omega}w_0{\rm d}x}{|\Omega|}, 0, \left(\frac{\lambda}{\mu}\right)^{\frac{1}{\kappa-1}}\right)$.

Key words: chemotaxis, signal-dependent motility, logistic source, boundedness, asymptotic behavior

中图分类号: 

  • 92C17