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Keywords:
Chemotaxis; signal-dependent sensitivity; logistic term; global existence.
Chemotaxis; signal-dependent sensitivity; logistic term; global existence.
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.
References:
[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. DOI 10.1142/S021820251550044X | MR 3351175
[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. DOI 10.3934/dcds.2015.35.1891 | MR 3294230
[3] Fujie, K.: Boundedness in a fully parabolic chemotaxis system with singular sensitivity. J.Math. Anal. Appl., 424, pp. 675–684, 2015. DOI 10.1016/j.jmaa.2014.11.045 | MR 3286587
[4] Fujie, K.: Study of reaction-diffusion systems modeling chemotaxis. PhD thesis, Tokyo University of Science, 2016.
[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. DOI 10.1088/0951-7715/29/8/2417 | MR 3538418
[6] Fujie, K., Senba, T.: A sufficient condition of sensitivity functions for boundedness of solutions to a parabolic-parabolic chemotaxis system. preprint. MR 3816648
[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. DOI 10.1016/j.jmaa.2015.12.058 | MR 3446989
[8] Horstmann, D., Wang, G.: Blow-up in a chemotaxis model without symmetry assumptions. Eur. J. Appl. Math., 12, pp. 159–177, 2001. DOI 10.1017/S0956792501004363 | MR 1931303
[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. DOI 10.1002/mma.3489 | MR 3454184
[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. DOI 10.1007/s00030-017-0472-8 | MR 3674184
[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
[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
[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. DOI 10.1016/j.jde.2016.05.008 | MR 3507983
[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
[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
[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. DOI 10.1137/140971853 | MR 3277217
[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. DOI 10.1016/j.jde.2014.11.009 | MR 3295594
[18] Winkler, M.: Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model. J. Differential Equations, 248, pp. 2889–2905, 2010. DOI 10.1016/j.jde.2010.02.008 | MR 2644137
[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. DOI 10.1016/j.jde.2014.04.023 | MR 3210023
[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. DOI 10.1016/j.jmaa.2014.03.084 | MR 3198865
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