Boundedness of solutions in a two-species chemotaxis system

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

In this paper, we consider an initial-Neumann boundary value problem for a two-species chemotaxis system

$\left\{\begin{array}{ll} \frac{\partial u}{\partial t}=\Delta u-\chi \nabla \cdot(u \nabla w)+u\left(a_{1}-b_{1} u^{m-1}+c_{1} v\right), & (x, t) \in \Omega \times\left(0, T_{\max }\right), \\ \frac{\partial v}{\partial t}=\Delta v-\xi \nabla \cdot(v \nabla w)+v\left(a_{2}-b_{2} v^{l-1}-c_{2} u\right), & (x, t) \in \Omega \times\left(0, T_{\max }\right), \\ \frac{\partial w}{\partial t}=\Delta w-\left(u^{\alpha}+v^{\beta}\right) w, & (x, t) \in \Omega \times\left(0, T_{\max }\right), \end{array}\right.$

where the domain $\Omega \subset \mathbb{R}^{n}$(n ≥ 2) is bounded and smooth,T_max ∈ (0, ∞], and parameters a_i,b_i,c_i,m,l, α, β, χ, ξ > 0 with m,l > 1, i = 1, 2. In the current work, we provide a sufficient condition of global classical solvability to the above system. More precisely, for some suitable initial data, if $m>\max \left\{\frac{\alpha(n+2)}{2}, 1\right\}$ and $l>\max \left\{\frac{\beta(n+2)}{2}, 1\right\}$, then the system has a global classical solution. Compared to previous work, the existence result established here is more generalized, depending only on the nonlinear power exponents and spatial dimensions.

Keywords

predator-prey model / classical solutions / global existence / nonlinear consumption

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] E. Keller, L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. https://doi.org/10.1016/0022-5193(70)90092-5

[2] C. Wang, P. Wang, X. Zhu, Global dynamics in a chemotaxis system involving nonlinear indirect signal secretion and logistic source, Z. Angew. Math. Phys., 74 (2023), 237. https://doi.org/10.1007/s00033-023-02126-2

[3] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Commun. Partial Differ. Equations, 35 (2010), 1516-1537. https://doi.org/10.1080/03605300903473426

[4] C. Lo, J. Rodrigues, Global existence for nonlocal quasilinear diffusion systems in nonisotropic nondivergence form, Math. Nachr., 297 (2024), 2122-2147. https://doi.org/10.1002/mana.202200250

[5] M. Winkler, Does a “volume-filling effect” always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24. https://doi.org/10.1002/mma.1146

[6] C. Wang, J. Zhu, Blow-up analysis to a quasilinear chemotaxis system with nonlocal logistic effect, Bull. Malays. Math. Sci. Soc., 47 (2024), 60. https://doi.org/10.1007/s40840-024-01659-7

[7] M. Winkler, How far do chemotaxis-driven forces influence regularity in the Navier-Stokes system?, Trans. Am. Math. Soc., 369 (2017), 3067-3125. https://doi.org/10.1090/tran/6733

[8] J. Zheng, An optimal result for global existence and boundedness in a three- dimensional Keller-Segel-Stokes, J. Differ. Equations, 267 (2019), 2385-2415. https://doi.org/10.1016/j.jde.2019.03.013

[9] W. Zhang, Global solutions in a chemotaxis consumption model with singular sensitivity, Z. Angew. Math. Phys., 74 (2023), 165. https://doi.org/10.1007/s00033-023-02049-y

[10] H. Liu, C. Lo, Determining a parabolic system by boundary observation of its non-negative solutions with biological applications, Inverse Probl., 40 (2024), 025009. https://doi.org/10.1088/1361-6420/ad149f

[11] N. Bellomo, A. Bellouquid, Y. Tao, M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763. https://doi.org/10.1142/S021820251550044X

[12] P. Liu, J. Shi, Z. Wang, Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2597-2625. https://doi.org/10.3934/dcdsb.2013.18.2597

[13] G. Fragnelli, Positive periodic solutions for a system of anisotropic parabolic equations, J. Math. Anal. Appl., 367 (2010), 204-228. https://doi.org/10.1016/j.jmaa.2009.12.039

[14] H. Alaa, N. Alaa, A. Bouchriti, A. Charkaoui, An improved nonlinear anisotropic model with p(x)-growth conditions applied to image restoration and enhancement, Math. Meth. Appl. Sci., 47 (2024), 7546-7575. https://doi.org/10.1002/mma.9989

[15] A. Charkaoui, A. Ben-Loghfyry, S. Zeng, A novel parabolic model driven by double phase flux operator with variable exponents: Application to image decomposition and denoising, Comput. Math. Appl., 174 (2024), 97-141. https://doi.org/10.1016/j.camwa.2024.08.021

[16] A. Charkaoui, A. Ben-Loghfyry, S. Zeng, Nonlinear parabolic double phase variable exponent systems with applications in image noise removal, Appl. Math. Modell., 132 (2024), 495-530. https://doi.org/10.1016/j.apm.2024.04.059

[17] J. Ahn, M. Winkler, A critical exponent for blow-up in a two-dimensional chemotaxis-consumption system, Calc. Var. Partial Differ. Equations, 62 (2023), 6. https://doi.org/10.1007/s00526-023-02523-5

[18] W. Lyu, Asymptotic stabilization for a class of chemotaxis-consumption systems with generalized logistic source, Nonlinear Anal., 217 (2022), 112737. https://doi.org/10.1016/j.na.2021.112737

[19] Y. Wang, M. Winkler, Finite-time blow-up in a repulsive chemotaxis-consumption system, Proc. Roy. Soc. Edinburgh Sect. A, 153 (2023), 1150-1166. https://doi.org/10.1017/prm.2022.39

[20] Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521-529. https://doi.org/10.1016/j.jmaa.2011.02.041

[21] K. Baghaei, A. Khelghati, Boundedness of classical solutions for a chemotaxis model with consumption of chemoattractant, C. R. Math. Acad. Sci. Paris, 355 (2017), 633-639. https://doi.org/10.1002/mma.4264

[22] M. Fuest, Analysis of a chemotaxis model with indirect signal absorption, J. Differ. Equations, 267 (2019), 4778-4806. https://doi.org/10.1016/j.jde.2019.05.015

[23] J. Lankeit, Y. Wang, Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption, Discrete Contin. Dyn. Syst., 37 (2017), 6099-6121. https://doi.org/10.3934/dcds.2017262

[24] C. Wang, Z. Zheng, X. Zhu, Dynamic behavior analysis to a generalized chemotaxis-consumption system, J. Math. Phys., 65 (2024), 011503. https://doi.org/10.1063/5.0176530

[25] Y. Chiyo, S. Frassu, G. Viglialoro, A nonlinear attraction-repulsion Keller-Segel model with double sublinear absorptions: Criteria toward boundedness, Commun. Pure Appl. Anal., 22 (2023), 1783-1809. https://doi.org/10.3934/cpaa.2023047

[26] C. Wang, Z. Zheng, Global boundedness for a chemotaxis system involving nonlinear indirect consumption mechanism, Discrete Contin. Dyn. Syst. B, 29 (2024), 2141-2157. https://doi.org/10.3934/dcdsb.2023171

[27] X. Bai, M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583. https://doi.org/10.1512/iumj.2016.65.5776

[28] M. Mizukami, Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2301-2319. https://doi.org/10.3934/dcdsb.2017097

[29] M. Mizukami, Improvement of conditions for asymptotic stability in a two-species chemotaxiscompetition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 13 (2020), 269-278. https://doi.org/10.3934/dcdss.2020015

[30] K. Lin, C. Mu, Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2233-2260. https://doi.org/10.3934/dcdsb.2017094

[31] K. Lin, C. Mu, L. Wang, Boundedness in a two-species chemotaxis system, Math. Methods Appl. Sci., 38 (2015), 5085-5096. https://doi.org/10.1002/mma.3429

[32] L. Miao, H. Yang, S. Fu, Global boundedness in a two-species predator-prey chemotaxis model, Appl. Math. Lett., 111 (2021), 106-639. https://doi.org/10.1016/j.aml.2020.106639

[33] L. Wang, C. Mu, X. Hu, P. Zheng, Boundedness and asymptotic stability of solutions to a two- species chemotaxis system with consumption of chemoattractant, J. Differ. Equations, 264 (2018), 3369-3401. https://doi.org/10.1016/j.jde.2017.11.019

[34] J. Zhang, X. Hu, L. Wang, L. Qu, Boundedness in a quasilinear two-species chemotaxis system with consumption of chemoattractant, Electron. J. Qual. Theory Differ. Equations, 31 (2019), 1-12. https://doi.org/10.14232/ejqtde.2019.1.31

[35] Q. Zhang, W. Tao, Boundedness and stabilization in a two-species chemotaxis system with signal absorption, Comput. Math. Appl., 78 (2019), 2672-2681. https://doi.org/10.1016/j.camwa.2019.04.008

[36] G. Ren, B. Liu, Global existence and asymptotic behavior in a two-species chemotaxis system with logistic source, J. Differ. Equations, 269 (2020), 1484-1520. https://doi.org/10.1016/j.jde.2020.01.008

[37] G. Ren, B. Liu, Global solvability and asymptotic behavior in a two-species chemotaxis system with Lotka-Volterra competitive kinetics, Math. Models Methods Appl. Sci., 31 (2021), 941-978. https://doi.org/10.1142/S0218202521500238

[38] M. Winkler, Global generalized solutions to a multi-dimensional doubly tactic resource consumption model accounting for social interactions, Math. Models Methods Appl. Sci., 29 (2019), 373-418. https://doi.org/10.1142/S021820251950012X

[39] Y. Tao, M. Winkler, Large time behavior in a forager-exploiter model with different taxis strategies for two groups in search of food, Math. Models Methods Appl. Sci., 29 (2019), 2151-2182. https://doi.org/10.1142/S021820251950043X

[40] J. Wang, M. Wang, Global bounded solution of the higher-dimensional forager-exploiter model with/without growth sources, Math. Models Methods Appl. Sci., 30 (2020), 1297-1323. https://doi.org/10.1142/S0218202520500232

[41] D. Wu, S. Shen, Global boundedness and stabilization in a forager-exploiter model with logistic growth and nonlinear resource consumption, Nonlinear Anal. Real World Appl., 72 (2023), 103-854. https://doi.org/10.1016/j.nonrwa.2023.103854

[42] H. Ou, L. Wang, Boundedness in a two-species chemotaxis system with nonlinear resource consumption, Qual. Theory Dyn. Syst., 23 (2024), 14. https://doi.org/10.1007/s12346-023-00873-1

[43] Y. Choi, Z. Wang, Prevention of blow-up by fast diffusion in chemotaxis, J. Math. Anal. Appl., 362 (2010), 553-564. https://doi.org/10.1016/j.jmaa.2009.08.012

[44] M. Delgado, I. Gayte, C. Morales-Rodrigo, A. Suárez, An angiogenesis model with nonlinear chemotactic response and flux at the tumor boundary, Nonlinear Anal., 72 (2010), 330-347. https://doi.org/10.1016/j.na.2009.06.057

[45] H. Schmeisser, H. Triebel, Function spaces, differential operators and nonlinear analysis, in Teubner-Texte zur Mathematik, 133 (1993), 9-126. https://doi.org/10.1007/978-3-663-11336-2

[46] J. Wang, Global existence and boundedness of a forager-exploiter system with nonlinear diffusions, J. Differ. Equations, 276 (2021), 460-492. https://doi.org/10.1016/j.jde.2020.12.028

[47] C. Stinner, C. Surulescu, M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007. https://doi.org/10.1137/13094058X

[48] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segal model, J. Differ. Equations, 248 (2010), 2889-2905. https://doi.org/10.1016/j.jde.2010.02.008

[49] J. Wang, Global existence and stabilization in a forager-exploiter model with general logistic sources, Nonlinear Anal., 222 (2022), 112-985. https://doi.org/10.1016/j.na.2022.112985

[50] A. Friedman, Partial Different Equations , Holt Rinehart and Winston, New York, 1969.

[51] L. Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa Sci. Fis. Mat., III. Ser., 13 (1959), 115-162.