Global boundedness for a chemotaxis-competition system with signal dependent sensitivity and loop

  • Received: 01 February 2021 Revised: 01 April 2021 Published: 26 May 2021
  • Primary: 35A01, 35K55; Secondary: 92C17

  • In this work, the fully parabolic chemotaxis-competition system with loop

    $ \begin{eqnarray*} \left\{ \begin{array}{llll} &\partial_{t} u_{1} = d_1\Delta u_{1}-\nabla\cdot(u_{1}\chi_{11}(v_{1})\nabla v_{1}) \\& \qquad -\nabla\cdot(u_{1}\chi_{12}(v_{2})\nabla v_{2}) +\mu_{1}u_{1}(1-u_{1}-a_{1}u_{2}),\\ &\partial_{t} u_{2} = d_2\Delta u_{2}-\nabla\cdot(u_{2}\chi_{21}(v_{1})\nabla v_{1}) \\& \qquad -\nabla\cdot(u_{2}\chi_{22}(v_{2})\nabla v_{2}) +\mu_{2}u_{2}(1-u_{2}-a_{2}u_{1}), \\ &\partial_t v_1 = d_3\Delta v_{1}-\lambda_{1} v_{1}+h_1(u_{1}, u_{2}), \\ &\partial_t v_2 = d_4\Delta v_{2}-\lambda_{2} v_{2}+h_2(u_{1}, u_{2}) \\ \end{array} \right. \end{eqnarray*} $

    is considered under the homogeneous Neumann boundary condition, where $ x\in\Omega, t>0 $, $ \Omega\subset \mathbb{R}^{n} (n\leq 3) $ is a bounded domain with smooth boundary. For any regular nonnegative initial data, it is proved that if the parameters $ \mu_1, \mu_2 $ are sufficiently large, then the system possesses a unique and global classical solution for $ n\leq 3 $. Specifically, when $ n = 2 $, the global boundedness can be attained without any constraints on $ \mu_1, \mu_2 $.

    Citation: Chun Huang. Global boundedness for a chemotaxis-competition system with signal dependent sensitivity and loop[J]. Electronic Research Archive, 2021, 29(5): 3261-3279. doi: 10.3934/era.2021037

    Related Papers:

  • In this work, the fully parabolic chemotaxis-competition system with loop

    $ \begin{eqnarray*} \left\{ \begin{array}{llll} &\partial_{t} u_{1} = d_1\Delta u_{1}-\nabla\cdot(u_{1}\chi_{11}(v_{1})\nabla v_{1}) \\& \qquad -\nabla\cdot(u_{1}\chi_{12}(v_{2})\nabla v_{2}) +\mu_{1}u_{1}(1-u_{1}-a_{1}u_{2}),\\ &\partial_{t} u_{2} = d_2\Delta u_{2}-\nabla\cdot(u_{2}\chi_{21}(v_{1})\nabla v_{1}) \\& \qquad -\nabla\cdot(u_{2}\chi_{22}(v_{2})\nabla v_{2}) +\mu_{2}u_{2}(1-u_{2}-a_{2}u_{1}), \\ &\partial_t v_1 = d_3\Delta v_{1}-\lambda_{1} v_{1}+h_1(u_{1}, u_{2}), \\ &\partial_t v_2 = d_4\Delta v_{2}-\lambda_{2} v_{2}+h_2(u_{1}, u_{2}) \\ \end{array} \right. \end{eqnarray*} $

    is considered under the homogeneous Neumann boundary condition, where $ x\in\Omega, t>0 $, $ \Omega\subset \mathbb{R}^{n} (n\leq 3) $ is a bounded domain with smooth boundary. For any regular nonnegative initial data, it is proved that if the parameters $ \mu_1, \mu_2 $ are sufficiently large, then the system possesses a unique and global classical solution for $ n\leq 3 $. Specifically, when $ n = 2 $, the global boundedness can be attained without any constraints on $ \mu_1, \mu_2 $.



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    [1] Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics. Indiana Univ. Math. J. (2016) 65: 553-583.
    [2] Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues. Math. Models Methods Appl. Sci. (2015) 25: 1663-1763.
    [3] Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals. Discrete Contin. Dyn. Syst. Ser. B (2017) 22: 1253-1272.
    [4] A simultaneous blow-up problem arising in tumor Modeling. J. Math. Biol. (2019) 79: 1357-1399.
    [5] Mathematical model of macrophage-facilitated breast cancer cells invasion. J. Theor. Biol. (2014) 357: 184-199.
    [6] Boundedness in a two-species chemotaxis parabolic system with two chemicals. Discrete Contin. Dyn. Syst. Ser. B (2017) 22: 2717-2729.
    [7] Nondegeneracy of blow-up points for the parabolic Keller-Segel system. Ann. Inst. H. Poincar'e Anal. Non Lin'eaire (2014) 31: 851-875.
    [8] X. Pan, L. Wang, J. Zhang and J. Wang, Boundedness in a three-dimensional two-species chemotaxis system with two chemicals, Z. Angew. Math. Phys., 71 (2020). doi: 10.1007/s00033-020-1248-2
    [9] Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion. SIAM J. Math. Anal. (2014) 46: 1969-2007.
    [10] Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant. J. Differential Equations (2014) 257: 784-815.
    [11] Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals. Discrete Contin. Dyn. Syst. Ser. B (2015) 20: 3165-3183.
    [12] Global dynamics in a two species chemotaxis competition system with two signals. Discrete Contin. Dyn. Syst. (2018) 38: 3617-3636.
    [13] X. Tu, C. Mu and S. Qiu, Global asymptotic stability in a parabolic-elliptic chemotaxis system with competitive kinetics and loop, Appl. Anal., (2020). doi: 10.1080/00036811.2020.1783536
    [14] X. Tu, C. Mu and S. Qiu, Boundedness and convergence of constant equilibria in a two-species chemotaxis-competition system with loop, Nonlinear Anal., 198 (2020), 111923. doi: 10.1016/j.na.2020.111923
    [15] X. Tu, C. Mu, S. Qiu and L. Yang, Boundedness in the higher-dimensional fully parabolic chemotaxis-competition system with loop, Z. Angew. Math. Phys., 71 (2020), 185. doi: 10.1007/s00033-020-01413-6
    [16] X. Tu, C.-L. Tang and S. Qiu, The phenomenon of large population densities in a chemotaxis-competition system with loop, J. Evol. Equ., (2020). doi: 10.1007/s00028-020-00650-6
    [17] L. Wang, J. Zhang C. Mu and X. Hu, Boundedness and stabilization in a two-species chemotaxis system with two Chemicals, Discrete Contin. Dyn. Syst. B, 25 (2020), 191-221. doi: 10.3934/dcdsb.2019178
    [18] L. Wang and C. Mu, A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. B, 25 (2020), 4585-4601. doi: 10.3934/dcdsb.2020114
    [19] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537. doi: 10.1080/03605300903473426
    [20] H. Yu, W. Wang and S. Zheng, Criteria on global boundedness versus finite time blow-up to a two-species chemotaxis system with two chemicals, Nonlinearity, 31 (2018), 502-514. doi: 10.1088/1361-6544/aa96c9
    [21] Q. Zhang, Competitive exclusion for a two-species chemotaxis system with two chemicals, Appl. Math. Lett., 83 (2018), 27-32. doi: 10.1016/j.aml.2018.03.012
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