Global boundedness and large time behavior of solutions to a chemotaxis–consumption system with signal-dependent motility

被引:0
作者
Dan Li
Jie Zhao
机构
[1] South China University of Technology,School of Mathematics
[2] China West Normal University,College of Mathematics and Information
来源
Zeitschrift für angewandte Mathematik und Physik | 2021年 / 72卷
关键词
Chemotaxis; Boundedness; Signal-dependent motility; 92C17; 35K55; 35Q92;
D O I
暂无
中图分类号
学科分类号
摘要
This paper deals with the following chemotaxis–consumption system with signal-dependent motility ut=Δ(γ(v)u),x∈Ω,t>0,vt=Δv-uv,x∈Ω,t>0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{llll} u_{t}=\Delta (\gamma (v)u),\,\,\, &{}x\in \Omega ,\,\,\, t>0,\\ v_{t}=\Delta v-uv, \,\,\,&{}x\in \Omega ,\,\,\, t>0,\\ \end{array} \right. \end{aligned}$$\end{document}under no-flux boundary conditions in a smoothly bounded domain Ω⊂Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset {\mathbb {R}}^{n}$$\end{document}. γ(s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma (s)$$\end{document} is the motility function. For the case of positive motility function, it is shown that the corresponding initial boundary value problem possesses a unique global classical solution which is uniformly bounded. Moreover, it is asserted that the solution to the system exponentially converges to constant equilibria in the large time. Finally, if the motility function is zero at some point, we obtain the existence of the weak solution.
引用
收藏
相关论文
共 63 条
  • [1] Ahn J(2019)Global well-posedness and stability of constant equilibria in parabolic–elliptic chemotaxis systems without gradient sensing Nonlinearity 32 1327-1351
  • [2] Yoon C(2012)Finite-time blowup and global-in-time unbounded solutions to a parabolic–parabolic quasilinear Keller–Segel system in higher dimensions J. Differ. Equ. 252 5832-5851
  • [3] Cieálak T(2020)Global existence for a kinetic model of pattern formation with density-suppressed motilities J. Differ. Equ. 269 5338-5378
  • [4] Stinner C(2002)Stability of solutions of chemotaxis equations in reinforced random walks J. Math. Anal. Appl. 272 138-163
  • [5] Fujie K(2001)Blow-up in a chemotaxis model without symmetry assumptions Eur. J. Appl. Math. 12 159-177
  • [6] Jiang J(2014)Boundedness in quasilinear Keller–Segel system of parabolic-parabolic type on nonconvex bounded domains J. Differ. Equ. 256 2993-3010
  • [7] Friedman A(2018)Boundedness, stabilization, and pattern formation driven by density-suppressed motility SIAM J. Appl. Math. 78 1632-1657
  • [8] Tello JI(2020)Boundedness and asymptotics of a reaction–diffusion system with density-dependent motility J. Differ. Equ. 269 6758-6793
  • [9] Horstmann D(1970)Initiation of slime mold aggregation viewed as an instability J. Theor. Biol. 26 399-415
  • [10] Wang G(2020)Global existence for a class of chemotaxis-consumption systems with signaldependent motility and generalized logistic source Nonlinear Anal. Real World Appl. 56 103160-601