Boundedness and asymptotic stability of solutions to a two-species chemotaxis system with consumption of chemoattractant. (English) Zbl 1380.35025
A two-competing-species chemotactic system is considered with logistic terms and consumption of chemoattractant in bounded domains with the homogeneous Neumann condition. Conditions guaranteeing global in time existence of solutions are given. Moreover, under some conditions, solutions are shown to stabilize to space-homogeneous steady states.
Reviewer: Piotr Biler (Wrocław)
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
| 35K58 | Semilinear parabolic equations |
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 35B35 | Stability in context of PDEs |
| 92C17 | Cell movement (chemotaxis, etc.) |
References:
| [1] | Alikakos, N. D., \(L^p\) bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4, 827-868 (1979) · Zbl 0421.35009 |
| [2] | Bai, X.; Winkler, M., Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65, 553-583 (2016) · Zbl 1345.35117 |
| [3] | Baghaeia, K.; Khelghatib, A., Global existence and boundedness of classical solutions for a chemotaxis model with consumption of chemoattractant and logistic source, Math. Methods Appl. Sci., 40, 3799-3807 (2017) · Zbl 1516.35010 |
| [4] | Black, T.; Lankeit, J.; Mizukami, M., On the weakly competitive case in a two-species chemotaxis model, IMA J. Appl. Math., 81, 860-876 (2016) · Zbl 1404.35033 |
| [5] | Cao, X.; Kurima, S.; Mizukami, M., Global existence and asymptotic behavior of classical solutions for a 3D two-species chemotaxis-Stokes system with competitive kinetics (2017) |
| [6] | Cao, X.; Kurima, S.; Mizukami, M., Global existence and asymptotic behavior of classical solutions for a 3D two-species Keller-Segel-Stokes system with competitive kinetics (2017) |
| [7] | Cieślak, T.; Winkler, M., Stabilization in a higher-dimensional quasilinear Keller-Segel system with exponentially decaying diffusivity and subcritical sensitivity, Nonlinear Anal., 159, 129-144 (2017) · Zbl 1367.35033 |
| [8] | Fasano, A.; Mancini, A.; Primicerio, M., Equilibrium of two populations subjected to chemotaxis, Math. Models Methods Appl. Sci., 14, 503-533 (2004) · Zbl 1090.34032 |
| [9] | Fan, L.; Jin, H., Global existence and asymptotic behavior to a chemotaxis system with consumption of chemoattractant in higher dimensions, J. Math. Phys., 58, Article 011503 pp. (2017) · Zbl 1355.76090 |
| [10] | Horstmann, D.; Winkler, M., Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215, 52-107 (2005) · Zbl 1085.35065 |
| [11] | Horstemann, D., Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci., 21, 231-270 (2011) · Zbl 1262.35203 |
| [12] | Hirata, M.; Kurima, S.; Mizukami, M.; Yokota, T., Boundedness and stabilization in a two-dimensional two-species chemotaxis-Navier-Stokes system with competitive kinetics, J. Differential Equations, 263, 470-490 (2017) · Zbl 1362.35049 |
| [13] | Hirata, M.; Kurima, S.; Mizukami, M.; Yokota, T., Boundedness and stabilization in a three-dimensional two-species chemotaxis-Navier-Stokes system with competitive kinetics (2017) · Zbl 1362.35049 |
| [14] | Jin, H.; Xiang, T., Convergence rates of solutions for a two-species chemotaxis-Navier-Stokes system with competitive kinetics (2017) |
| [15] | Keller, E. F.; Segel, L. A., Traveling bands of chemotactic bacteria: a theoretical analysis, J. Theoret. Biol., 30, 377-380 (1971) · Zbl 1170.92308 |
| [16] | Kowalczyk, R.; Szymańska, Z., On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343, 379-398 (2008) · Zbl 1143.35333 |
| [17] | Lin, K.; Mu, C.; Wang, L., Boundedness in a two-species chemotaxis system, Math. Methods Appl. Sci., 38, 5085-5096 (2015) · Zbl 1334.92059 |
| [18] | Lin, K.; Mu, C., Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source, Discrete Contin. Dyn. Syst. Ser. B, 22, 2233-2260 (2017) · Zbl 1366.35067 |
| [19] | Lankeit, J.; Wang, Y., Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption (2016) |
| [20] | Mizukami, M., Remarks on smallness of chemotactic effect for asymptotic stability in a two-species chemotaxis system, AIMS Math., 1, 156-164 (2016) · Zbl 1435.35393 |
| [21] | 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, 2650-2669 (2016) · Zbl 1339.35197 |
| [22] | Mizukami, M., Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 22, 2301-2319 (2017) · Zbl 1366.35068 |
| [23] | Mizukami, M., Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity (2017) · Zbl 1366.35068 |
| [24] | Mu, C.; Wang, L.; Zheng, P.; Zhang, Q., Global existence and boundedness of classical solutions to a parabolic-parabolic chemotaxis system, Nonlinear Anal. Real World Appl., 14, 1634-1642 (2013) · Zbl 1261.35072 |
| [25] | Negreanu, M.; Tello, J. I., On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal., 46, 3761-3781 (2014) · Zbl 1317.35109 |
| [26] | Negreanu, M.; Tello, J. I., Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant, J. Differential Equations, 258, 1592-1617 (2015) · Zbl 1310.35040 |
| [27] | Porzio, M. M.; Vespri, V., Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103, 146-178 (1993) · Zbl 0796.35089 |
| [28] | Stinner, C.; Tello, J. I.; Winkler, M., Competitive exclusion in a two-species chemotaxis model, J. Math. Biol., 68, 1607-1626 (2014) · Zbl 1319.92050 |
| [29] | Tao, Y., Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381, 521-529 (2011) · Zbl 1225.35118 |
| [30] | Tao, Y.; Winkler, M., Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differential Equations, 252, 2520-2543 (2012) · Zbl 1268.35016 |
| [31] | Tao, Y.; Winkler, M., Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252, 692-715 (2012) · Zbl 1382.35127 |
| [32] | Tao, Y.; Winkler, M., Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47, 4229-4250 (2015) · Zbl 1328.35103 |
| [33] | Tello, J. I.; Winkler, M., Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25, 1413-1425 (2012) · Zbl 1260.92014 |
| [34] | Tuval, I.; Cisneros, L.; Dombrowski, C.; Wolgemuth, C. W.; Kessler, J. O.; Goldstein, R. E., Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA, 102, 2277-2282 (2005) · Zbl 1277.35332 |
| [35] | Wang, L.; Mu, C.; Zhou, S., Boundedness in a parabolic-parabolic chemotaxis system with nonlinear diffusion, Z. Angew. Math. Phys., 65, 1137-1152 (2014) · Zbl 1305.92024 |
| [36] | Wang, L.; Mu, C.; Lin, K.; Zhao, J., Global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant, Z. Angew. Math. Phys., 66, 1633-1648 (2015) · Zbl 1328.92019 |
| [37] | Wang, L.; Mu, C.; Hu, X., Global solutions to a chemotaxis model with consumption of chemoattractant, Z. Angew. Math. Phys., 288 (2016) · Zbl 1381.92013 |
| [38] | Wang, L.; Hu, X.; Zheng, P.; Li, L., Boundedness in a chemotaxis model with exponentially decaying diffusivity and consumption of chemoattractant, Comput. Math. Appl., 74, 10, 2444-2448 (2017) · Zbl 1396.92011 |
| [39] | Wang, L.; Mu, C.; Hu, X.; Zheng, Pan, Boundedness in a quasilinear chemotaxis model with consumption of chemoattractant and logistic source, Appl. Anal. (2017) |
| [40] | Wang, Y.; Xiang, Z., Global existence and boundedness in a higher-dimensional quasilinear chemotaxis system, Z. Angew. Math. Phys., 66, 3159-3179 (2015) · Zbl 1330.35201 |
| [41] | Winkler, M., Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr., 283, 1664-1673 (2010) · Zbl 1205.35037 |
| [42] | Winkler, M., Large-data global generalized solutions in a chemotaxis system with tensor-valued sensitivities, SIAM J. Math. Anal., 47, 3092-3115 (2015) · Zbl 1330.35202 |
| [43] | Winkler, M., Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37, 319-352 (2012) · Zbl 1236.35192 |
| [44] | Winkler, M., Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35, 1516-1537 (2010) · Zbl 1290.35139 |
| [45] | Wolansky, G., Multi-components chemotaxis system in absence of conflict, European J. Appl. Math., 13, 641-661 (2002) · Zbl 1008.92008 |
| [46] | Zhang, Q.; Li, Y., Stabilization and convergence rate in a chemotaxis system with consumption of chemoattractant, J. Math. Phys., 56, Article 081506 pp. (2015) · Zbl 1322.35060 |
| [47] | Zhang, Q.; Li, Y., Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys., 66, 83-93 (2002) · Zbl 1333.35097 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
