Title: | Boundedness in a fully parabolic chemotaxis system with signal-dependent sensitivity and logistic term (English) |

Author: | Mizukami, Masaaki |

Language: | English |

Journal: | Proceedings of Equadiff 14 |

Volume: | Conference on Differential Equations and Their Applications, Bratislava, July 24-28, 2017 |

Issue: | 2017 |

Year: | |

Pages: | 61-68 |

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Category: | math |

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Summary: | This paper deals with the chemotaxis system with signal-dependent sensitivity and logistic term \begin{align*} &u_t=\Delta u - \nabla \cdot (u \chi(v)\nabla v) + \mu u(1-u), % \quad \\ &v_t=\Delta v + u - v \end{align*} in $\Omega\times (0,\infty)$, where $\Omega$ is a bounded domain in $\mathbb{R}^n$ ($n\ge 2$) with smooth boundary, $\mu > 0$ is a constant and $\chi$ is a function generalizing \begin{align*} \chi(s) = \frac{K}{(1+s)^2} \quad (K>0,\ s>0). \end{align*} In the case that $\mu=0$ global existence and boundedness were established under some conditions for global existence and boundedness in the above system.([14]); however, conditions for global existence and boundedness in the above system have not been studied. The purpose of this paper is to construct conditions for global existence and boundedness in the above system. (English) |

Keyword: | Chemotaxis; signal-dependent sensitivity; logistic term; global existence. |

MSC: | 35A01 |

MSC: | 35K51 |

MSC: | 92C17. |

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Date available: | 2019-09-27T07:39:40Z |

Last updated: | 2019-09-27 |

Stable URL: | http://hdl.handle.net/10338.dmlcz/703018 |

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Reference: | [1] Bellomo, N., Bellouquid, A., Tao, Y., Winkler, M.: Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues., Math. Models Methods Appl. Sci., 25, pp. 1663–1763, 2015. MR 3351175, 10.1142/S021820251550044X |

Reference: | [2] Cao, X.: Global bounded solutions of the higher-dimensional Keller–Segel system under smallness conditions in optimal spaces., Discrete Contin. Dyn. Syst., 35, pp. 1891–1904, 2015. MR 3294230, 10.3934/dcds.2015.35.1891 |

Reference: | [3] Fujie, K.: Boundedness in a fully parabolic chemotaxis system with singular sensitivity., J.Math. Anal. Appl., 424, pp. 675–684, 2015. MR 3286587, 10.1016/j.jmaa.2014.11.045 |

Reference: | [4] Fujie, K.: Study of reaction-diffusion systems modeling chemotaxis., PhD thesis, Tokyo University of Science, 2016. |

Reference: | [5] Fujie, K., Senba, T.: Global existence and boundedness of radial solutions to a two dimensional fully parabolic chemotaxis system with general sensitivity., Nonlinearity, 29, pp. 2417–2450, 2016. MR 3538418, 10.1088/0951-7715/29/8/2417 |

Reference: | [6] Fujie, K., Senba, T.: A sufficient condition of sensitivity functions for boundedness of solutions to a parabolic-parabolic chemotaxis system., preprint. MR 3816648 |

Reference: | [7] He, X., Zheng, S.: Convergence rate estimates of solutions in a higher dimensional chemotaxis system with logistic source., J. Math. Anal. Appl., 436, pp. 970–982, 2016. MR 3446989, 10.1016/j.jmaa.2015.12.058 |

Reference: | [8] Horstmann, D., Wang, G.: Blow-up in a chemotaxis model without symmetry assumptions., Eur. J. Appl. Math., 12, pp. 159–177, 2001. MR 1931303, 10.1017/S0956792501004363 |

Reference: | [9] Lankeit, J.: A new approach toward boundedness in a two-dimensional parabolic chemotaxis system with singular sensitivity., Math. Methods Appl. Sci., 39, pp. 394–404, 2016. MR 3454184, 10.1002/mma.3489 |

Reference: | [10] Lankeit, J., Winkler, M.: A generalized solution concept for the Keller–Segel system with logarithmic sensitivity: global solvability for large nonradial data., NoDEA, Nonlinear Differ. Equ. Appl., 24, No. 4, Paper No. 49, 33 p., 2017. MR 3674184, 10.1007/s00030-017-0472-8 |

Reference: | [11] Mizukami, M.: Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity., Discrete Contin. Dyn. Syst. Ser. B, 22, pp. 2301–2319, 2017. MR 3664704 |

Reference: | [12] Mizukami, M.: Improvement of conditions for asymptotic stability in a two-species chemotaxis competition model with signal-dependent sensitivity., submitted, arXiv:1706.04774[math.AP]. MR 3664704 |

Reference: | [13] Mizukami, M., Yokota, T.: Global existence and asymptotic stability of solutions to a two-species chemotaxis system with any chemical diffusion., J. Differential Equations, 261, pp. 2650–2669, 2016. MR 3507983, 10.1016/j.jde.2016.05.008 |

Reference: | [14] Mizukami, M., Yokota, T.: A unified method for boundedness in fully parabolic chemotaxis systems with signal-dependent sensitivity., Math. Nachr., to appear. MR 3722501 |

Reference: | [15] Nagai, T., Senba, T., Yoshida, K.: Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis., Funkcial. Ekvac., 40, pp. 411–433, 1997. MR 1610709 |

Reference: | [16] Negreanu, M., Tello, J. I.: On a two species chemotaxis model with slow chemical diffusion., SIAM J. Math. Anal., 46, pp. 3761–3781, 2014. MR 3277217, 10.1137/140971853 |

Reference: | [17] Negreanu, M., Tello, J. I.: Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant., J. Differential Equations, 258, pp. 1592–1617, 2015. MR 3295594, 10.1016/j.jde.2014.11.009 |

Reference: | [18] Winkler, M.: Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model., J. Differential Equations, 248, pp. 2889–2905, 2010. MR 2644137, 10.1016/j.jde.2010.02.008 |

Reference: | [19] Winkler, M.: Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening., J. Differential Equations, 257, pp. 1056–1077, 2014. MR 3210023, 10.1016/j.jde.2014.04.023 |

Reference: | [20] Zhang, Q., Li, X.: Global existence and asymptotic properties of the solution to a two-species chemotaxis system., J. Math. Anal. Appl., 418, pp. 47–63, 2014. MR 3198865, 10.1016/j.jmaa.2014.03.084 |

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