Solutions of the attraction-repulsion-chemotaxis system with nonlinear diffusion
Abstract
In this study, we consider the well-posedness of the attraction-repulsion chemotaxis system. This paper explores the dynamics of species movement in reaction to two chemically opposing substances, incorporating nonlinear diffusion. Our primary objective is to establish the existence of a global-in-time weak solution for the proposed model in an unbounded three-dimensional spatial domain. Our study has confirmed the existence of a global-in-time weak solution for the proposed system in three dimensions. Furthermore, we demonstrate that global-in-time weak solutions are also attainable for the proposed system in a bounded domain with a smooth boundary.
Keyword : attraction-repulsion, chemotaxis, weak solutions, nonlinear diffusion
How to Cite
Karuppusamy, Y., & Lingeshwaran, S. (2025). Solutions of the attraction-repulsion-chemotaxis system with nonlinear diffusion. Mathematical Modelling and Analysis, 30(2), 203–223. https://doi.org/10.3846/mma.2025.19654
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
Y. Cai and Z. Li. A quasilinear attraction–repulsion chemotaxis system with logistic source. Nonlinear Analysis: Real World Applications, 70:103796, 2023. https://doi.org/10.1016/j.nonrwa.2022.103796
M. Chae, K. Kang and J. Lee. Existence of smooth solutions to coupled chemotaxis-fluid equations. Discrete and Continuous Dynamical Systems, 33(6):2271–2297, 2013. https://doi.org/10.3934/dcds.2013.33.2271
Y. Chiyo, M. Marras, Y. Tanaka and T. Yokota. Blow-up phenomena in a parabolic–elliptic–elliptic attraction–repulsion chemotaxis system with superlinear logistic degradation. Nonlinear Analysis, 212:112550, 2021. https://doi.org/10.1016/j.na.2021.112550
Y. Chiyo, M. Mizukami and T. Yokota. Global existence and boundedness in a fully parabolic attraction-repulsion chemotaxis system with signal-dependent sensitivities and logistic source. Journal of Mathematical Analysis and Applications, 489(1):124153, 2020. https://doi.org/10.1016/j.jmaa.2020.124153
Y.S. Chung and K. Kang. Existence of global solutions for a chemotaxisfluid system with nonlinear diffusion. J. Math. Phys., 57(4):041503, 2016. https://doi.org/10.1063/1.4947107
Y.S. Chung, K. Kang and J. Kim. Global existence of weak solutions for a Keller-Segel-Fluid model with nonlinear diffusion. Journal of the Korean Mathematical Society, 51(3):635–654, 2014. https://doi.org/10.4134/JKMS.2014.51.3.635
E. Espejo and T. Suzuki. Global existence and blow-up for a system describing the aggregation of microglia. Applied Mathematics Letters, 35:29–34, 2014. https://doi.org/10.1016/j.aml.2014.04.007
M.D. Francesco, A. Lorz and P.A. Markowich. Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior. Discrete and Continuous Dynamical Systems, 28(4):1437–1453, 2010. https://doi.org/10.3934/dcds.2010.28.1437
K. Fujie and T. Senba. Global existence and boundedness of radial solutions to a two dimensional fully parabolic chemotaxis system with general sensitivity. Nonlinearity, 29(8):2417, 2016. https://doi.org/10.1088/0951-7715/29/8/2417
X. He, M. Tian and S. Zheng. Large time behavior of solutions to a quasilinear attraction–repulsion chemotaxis system with logistic source. Nonlinear Analysis: Real World Applications, 54:103095, 2020. https://doi.org/10.1016/j.nonrwa.2020.103095
M.A. Herrero and J.J.L. Velázquez. Singularity patterns in a chemotaxis model. Math. Ann., 306(3):583–624, 1996. https://doi.org/10.1007/BF01445268
C.Y. Hsieh and Y. Yu. Boundedness of solutions to an attraction-repulsion chemotaxis model in R2. Journal of Differential Equations, 317:422–438, 2022. https://doi.org/10.1016/j.jde.2022.02.017
S. Ishida and T. Yokota. Global existence of weak solutions to quasilinear degenerate Keller–Segel systems of parabolic–parabolic type with small data. Journal of Differential Equations, 252(3):2469–2491, 2012. https://doi.org/10.1016/j.jde.2011.08.047
K. Kang and A. Stevens. Blowup and global solutions in a chemotaxis–growth system. Nonlinear Analysis: Theory, Methods & Applications, 135:57–72, 2016. https://doi.org/10.1016/j.na.2016.01.017
E.F. Keller and L.A. Segel. Model for chemotaxis. Journal of Theoretical Biology, 30(2):225–234, 1971. https://doi.org/10.1016/0022-5193(71)90050-6
J. Lankeit. Finite-time blow-up in the three-dimensional fully parabolic attraction-dominated attraction-repulsion chemotaxis system. Journal of Mathematical Analysis and Applications, 504(2):125409, 2021. https://doi.org/10.1016/j.jmaa.2021.125409
J. Liu and Z.A. Wang. Classical solutions and steady states of an attraction–repulsion chemotaxis in one dimension. Journal of Biological Dynamics, 6(sup1):31–41, 2012. https://doi.org/10.1080/17513758.2011.571722
J.G. Liu and A. Lorz. A coupled chemotaxis-fluid model: Global existence. Ann. l’Institut Henri Poincare Anal. Non Lineaire, 28(5):643–652, 2011. https://doi.org/10.1016/j.anihpc.2011.04.005
M. Liu and Y. Wang. Finite-time blow-up in the Cauchy problem of a Keller-Segel system with logistic source. Discrete Continuous Dyn. Syst. Ser. B, 28(10):5396–5417, 2023. https://doi.org/10.3934/dcdsb.2023075
M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner. Chemotactic signaling, microglia, and Alzheimer’s disease senile plaques: Is there a connection? Bull. Math. Biol., 65(4):693–730, 2003. https://doi.org/10.1016/S0092-8240(03)00030-2
M. Mimura and T. Tsujikawa. Aggregating pattern dynamics in a chemotaxis model including growth. Phys. A: Stat. Mech. Appl., 230(3):499–543, 1996. https://doi.org/10.1016/0378-4371(96)00051-9
T. Nagai. Global existence of solutions to a parabolic system for chemotaxis in two space dimensions. Nonlinear Anal. Theory. Methods Appl., 30(8):5381–5388, 1997. https://doi.org/10.1016/S0362-546X(97)00395-7
T. Nagai, Y. Seki, and T. Yamada. Boundedness of solutions to a parabolic attraction–repulsion chemotaxis system in R2: The attractive dominant case. Applied Mathematics Letters, 121:107354, 2021. https://doi.org/10.1016/j.aml.2021.107354
E. Nakaguchi and K. Osaki. Global existence of solutions to a parabolic–parabolic system for chemotaxis with weak degradation. Nonlinear Anal. Theory. Methods Appl., 74(1):286–297, 2011. https://doi.org/10.1016/j.na.2010.08.044
K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura. Exponential attractor for a chemotaxis-growth system of equations. Nonlinear Anal. Theory. Methods Appl., 51(1):119–144, 2002. https://doi.org/10.1016/S0362-546X(01)00815-X
Y. Sugiyama and H. Kunii. Global existence and decay properties for a degenerate Keller–Segel model with a power factor in drift term. Journal of Differential Equations, 227(1):333–364, 2006. https://doi.org/10.1016/j.jde.2006.03.003
Y. Tao and Z.-A. Wang. Competing effects of attraction vs. Repulsion in chemotaxis. Math. Models Methods Appl. Sci., 23(01):1–36, 2013. https://doi.org/10.1142/S0218202512500443
J.I. Tello and M. Winkler. A Chemotaxis System with Logistic Source. Communications in Partial Differential Equations, 32(6):849–877, 2007. https://doi.org/10.1080/03605300701319003
M. Winkler. Chemotaxis with logistic source: Very weak global solutions and their boundedness properties. Journal of Mathematical Analysis and Applications, 348(2):708–729, 2008. https://doi.org/10.1016/j.jmaa.2008.07.071
M. Winkler. How Far Can Chemotactic Cross-diffusion Enforce Exceeding Carrying Capacities? J. Nonlinear Sci., 24(5):809–855, 2014. https://doi.org/10.1007/s00332-014-9205-x
T. Xiang. Boundedness and global existence in the higherdimensional parabolic–parabolic chemotaxis system with/without growth source. Journal of Differential Equations, 258(12):4275–4323, 2015. https://doi.org/10.1016/j.jde.2015.01.032
T. Yamada. Global existence and boundedness of solutions to a parabolic attraction-repulsion chemotaxis system in R2: The repulsive dominant case. Journal of Differential Equations, 315:254–269, 2022. https://doi.org/10.1016/j.jde.2022.01.042
M. Chae, K. Kang and J. Lee. Existence of smooth solutions to coupled chemotaxis-fluid equations. Discrete and Continuous Dynamical Systems, 33(6):2271–2297, 2013. https://doi.org/10.3934/dcds.2013.33.2271
Y. Chiyo, M. Marras, Y. Tanaka and T. Yokota. Blow-up phenomena in a parabolic–elliptic–elliptic attraction–repulsion chemotaxis system with superlinear logistic degradation. Nonlinear Analysis, 212:112550, 2021. https://doi.org/10.1016/j.na.2021.112550
Y. Chiyo, M. Mizukami and T. Yokota. Global existence and boundedness in a fully parabolic attraction-repulsion chemotaxis system with signal-dependent sensitivities and logistic source. Journal of Mathematical Analysis and Applications, 489(1):124153, 2020. https://doi.org/10.1016/j.jmaa.2020.124153
Y.S. Chung and K. Kang. Existence of global solutions for a chemotaxisfluid system with nonlinear diffusion. J. Math. Phys., 57(4):041503, 2016. https://doi.org/10.1063/1.4947107
Y.S. Chung, K. Kang and J. Kim. Global existence of weak solutions for a Keller-Segel-Fluid model with nonlinear diffusion. Journal of the Korean Mathematical Society, 51(3):635–654, 2014. https://doi.org/10.4134/JKMS.2014.51.3.635
E. Espejo and T. Suzuki. Global existence and blow-up for a system describing the aggregation of microglia. Applied Mathematics Letters, 35:29–34, 2014. https://doi.org/10.1016/j.aml.2014.04.007
M.D. Francesco, A. Lorz and P.A. Markowich. Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior. Discrete and Continuous Dynamical Systems, 28(4):1437–1453, 2010. https://doi.org/10.3934/dcds.2010.28.1437
K. Fujie and T. Senba. Global existence and boundedness of radial solutions to a two dimensional fully parabolic chemotaxis system with general sensitivity. Nonlinearity, 29(8):2417, 2016. https://doi.org/10.1088/0951-7715/29/8/2417
X. He, M. Tian and S. Zheng. Large time behavior of solutions to a quasilinear attraction–repulsion chemotaxis system with logistic source. Nonlinear Analysis: Real World Applications, 54:103095, 2020. https://doi.org/10.1016/j.nonrwa.2020.103095
M.A. Herrero and J.J.L. Velázquez. Singularity patterns in a chemotaxis model. Math. Ann., 306(3):583–624, 1996. https://doi.org/10.1007/BF01445268
C.Y. Hsieh and Y. Yu. Boundedness of solutions to an attraction-repulsion chemotaxis model in R2. Journal of Differential Equations, 317:422–438, 2022. https://doi.org/10.1016/j.jde.2022.02.017
S. Ishida and T. Yokota. Global existence of weak solutions to quasilinear degenerate Keller–Segel systems of parabolic–parabolic type with small data. Journal of Differential Equations, 252(3):2469–2491, 2012. https://doi.org/10.1016/j.jde.2011.08.047
K. Kang and A. Stevens. Blowup and global solutions in a chemotaxis–growth system. Nonlinear Analysis: Theory, Methods & Applications, 135:57–72, 2016. https://doi.org/10.1016/j.na.2016.01.017
E.F. Keller and L.A. Segel. Model for chemotaxis. Journal of Theoretical Biology, 30(2):225–234, 1971. https://doi.org/10.1016/0022-5193(71)90050-6
J. Lankeit. Finite-time blow-up in the three-dimensional fully parabolic attraction-dominated attraction-repulsion chemotaxis system. Journal of Mathematical Analysis and Applications, 504(2):125409, 2021. https://doi.org/10.1016/j.jmaa.2021.125409
J. Liu and Z.A. Wang. Classical solutions and steady states of an attraction–repulsion chemotaxis in one dimension. Journal of Biological Dynamics, 6(sup1):31–41, 2012. https://doi.org/10.1080/17513758.2011.571722
J.G. Liu and A. Lorz. A coupled chemotaxis-fluid model: Global existence. Ann. l’Institut Henri Poincare Anal. Non Lineaire, 28(5):643–652, 2011. https://doi.org/10.1016/j.anihpc.2011.04.005
M. Liu and Y. Wang. Finite-time blow-up in the Cauchy problem of a Keller-Segel system with logistic source. Discrete Continuous Dyn. Syst. Ser. B, 28(10):5396–5417, 2023. https://doi.org/10.3934/dcdsb.2023075
M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner. Chemotactic signaling, microglia, and Alzheimer’s disease senile plaques: Is there a connection? Bull. Math. Biol., 65(4):693–730, 2003. https://doi.org/10.1016/S0092-8240(03)00030-2
M. Mimura and T. Tsujikawa. Aggregating pattern dynamics in a chemotaxis model including growth. Phys. A: Stat. Mech. Appl., 230(3):499–543, 1996. https://doi.org/10.1016/0378-4371(96)00051-9
T. Nagai. Global existence of solutions to a parabolic system for chemotaxis in two space dimensions. Nonlinear Anal. Theory. Methods Appl., 30(8):5381–5388, 1997. https://doi.org/10.1016/S0362-546X(97)00395-7
T. Nagai, Y. Seki, and T. Yamada. Boundedness of solutions to a parabolic attraction–repulsion chemotaxis system in R2: The attractive dominant case. Applied Mathematics Letters, 121:107354, 2021. https://doi.org/10.1016/j.aml.2021.107354
E. Nakaguchi and K. Osaki. Global existence of solutions to a parabolic–parabolic system for chemotaxis with weak degradation. Nonlinear Anal. Theory. Methods Appl., 74(1):286–297, 2011. https://doi.org/10.1016/j.na.2010.08.044
K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura. Exponential attractor for a chemotaxis-growth system of equations. Nonlinear Anal. Theory. Methods Appl., 51(1):119–144, 2002. https://doi.org/10.1016/S0362-546X(01)00815-X
Y. Sugiyama and H. Kunii. Global existence and decay properties for a degenerate Keller–Segel model with a power factor in drift term. Journal of Differential Equations, 227(1):333–364, 2006. https://doi.org/10.1016/j.jde.2006.03.003
Y. Tao and Z.-A. Wang. Competing effects of attraction vs. Repulsion in chemotaxis. Math. Models Methods Appl. Sci., 23(01):1–36, 2013. https://doi.org/10.1142/S0218202512500443
J.I. Tello and M. Winkler. A Chemotaxis System with Logistic Source. Communications in Partial Differential Equations, 32(6):849–877, 2007. https://doi.org/10.1080/03605300701319003
M. Winkler. Chemotaxis with logistic source: Very weak global solutions and their boundedness properties. Journal of Mathematical Analysis and Applications, 348(2):708–729, 2008. https://doi.org/10.1016/j.jmaa.2008.07.071
M. Winkler. How Far Can Chemotactic Cross-diffusion Enforce Exceeding Carrying Capacities? J. Nonlinear Sci., 24(5):809–855, 2014. https://doi.org/10.1007/s00332-014-9205-x
T. Xiang. Boundedness and global existence in the higherdimensional parabolic–parabolic chemotaxis system with/without growth source. Journal of Differential Equations, 258(12):4275–4323, 2015. https://doi.org/10.1016/j.jde.2015.01.032
T. Yamada. Global existence and boundedness of solutions to a parabolic attraction-repulsion chemotaxis system in R2: The repulsive dominant case. Journal of Differential Equations, 315:254–269, 2022. https://doi.org/10.1016/j.jde.2022.01.042