American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity
  • CPAA Home
  • This Issue
  • Next Article
    Dynamics in a diffusive predator-prey system with stage structure and strong allee effect
February  2020, 19(2): 911-922. doi: 10.3934/cpaa.2020041

Asymptotic profile of solutions to a certain chemotaxis system

Rafał Celiński and  Andrzej Raczyński ,

Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, POLAND

* Corresponding author

Received  December 2018 Revised  June 2019 Published  October 2019

We consider a Cauchy problem for a two-dimensional model of chemotaxis and we show that large time behavior of solution is given by a multiple of the heat kernel.

Keywords: Chemotaxis, Keller-Segel system, global existence, decay estimates.
Mathematics Subject Classification: Primary: 92C17, 35B40; Secondary: 35K55, 35Q92.
Citation: Rafał Celiński, Andrzej Raczyński. Asymptotic profile of solutions to a certain chemotaxis system. Communications on Pure & Applied Analysis, 2020, 19 (2) : 911-922. doi: 10.3934/cpaa.2020041
References:
[1]

P. Biler, M. Guedda and G. Karch, Asymptotic properties of solutions of the viscous Hamilton-Jacobi equation, J. Evol. Equ., 4 (2004), 75-97.  doi: 10.1007/s00028-003-0079-x.  Google Scholar

[2]

M. Chae, K. Kang and J. Lee, Global existence and temporal decay in Keller-Segel models coupled to fluid equations, Comm. Partial Differential Equations, 39 (2014), 1205-1235.  doi: 10.1080/03605302.2013.852224.  Google Scholar

[3]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[4]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[5]

H. Kozono, M. Miura and Y. Sugiyama, Existence and uniqueness theorem on mild solutions to the Keller-Segel system coupled with the Navier-Stokes fluid, J. Funct. Anal., 270 (2016), 1663-1683.  doi: 10.1016/j.jfa.2015.10.016.  Google Scholar

[6]

Y. Li and Y. Li, Global boundedness of solutions for the chemotaxis-Navier-Stokes system in $ \mathbb{R}^2$, J. Differential Equations, 261 (2016), 6570-6613.  doi: 10.1016/j.jde.2016.08.045.  Google Scholar

[7]

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

[8]

Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differential Equations, 252 (2012), 2520-2543.  doi: 10.1016/j.jde.2011.07.010.  Google Scholar

[9]

I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Nat. Acad. Sci. USA, 102 (2005), 2277-2282.  doi: 10.1073/pnas.0406724102.  Google Scholar

[10]

M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319-351.  doi: 10.1080/03605302.2011.591865.  Google Scholar

[11]

M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Ration. Mech. Anal., 211 (2014), 455-487.  doi: 10.1007/s00205-013-0678-9.  Google Scholar

[12]

Q. Zhang and X. Zheng, Global well-posedness for the two-dimensional incompressible chemotaxis-Navier-Stokes equations, SIAM J. Math. Anal., 46 (2014), 3078-3105.  doi: 10.1137/130936920.  Google Scholar

show all references

References:
[1]

P. Biler, M. Guedda and G. Karch, Asymptotic properties of solutions of the viscous Hamilton-Jacobi equation, J. Evol. Equ., 4 (2004), 75-97.  doi: 10.1007/s00028-003-0079-x.  Google Scholar

[2]

M. Chae, K. Kang and J. Lee, Global existence and temporal decay in Keller-Segel models coupled to fluid equations, Comm. Partial Differential Equations, 39 (2014), 1205-1235.  doi: 10.1080/03605302.2013.852224.  Google Scholar

[3]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[4]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[5]

H. Kozono, M. Miura and Y. Sugiyama, Existence and uniqueness theorem on mild solutions to the Keller-Segel system coupled with the Navier-Stokes fluid, J. Funct. Anal., 270 (2016), 1663-1683.  doi: 10.1016/j.jfa.2015.10.016.  Google Scholar

[6]

Y. Li and Y. Li, Global boundedness of solutions for the chemotaxis-Navier-Stokes system in $ \mathbb{R}^2$, J. Differential Equations, 261 (2016), 6570-6613.  doi: 10.1016/j.jde.2016.08.045.  Google Scholar

[7]

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

[8]

Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differential Equations, 252 (2012), 2520-2543.  doi: 10.1016/j.jde.2011.07.010.  Google Scholar

[9]

I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Nat. Acad. Sci. USA, 102 (2005), 2277-2282.  doi: 10.1073/pnas.0406724102.  Google Scholar

[10]

M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319-351.  doi: 10.1080/03605302.2011.591865.  Google Scholar

[11]

M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Ration. Mech. Anal., 211 (2014), 455-487.  doi: 10.1007/s00205-013-0678-9.  Google Scholar

[12]

Q. Zhang and X. Zheng, Global well-posedness for the two-dimensional incompressible chemotaxis-Navier-Stokes equations, SIAM J. Math. Anal., 46 (2014), 3078-3105.  doi: 10.1137/130936920.  Google Scholar

[1]

Qi Wang. Boundary spikes of a Keller-Segel chemotaxis system with saturated logarithmic sensitivity. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1231-1250. doi: 10.3934/dcdsb.2015.20.1231
[2]

Kentarou Fujie, Takasi Senba. Global existence and boundedness in a parabolic-elliptic Keller-Segel system with general sensitivity. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 81-102. doi: 10.3934/dcdsb.2016.21.81
[3]

Chao Deng, Tong Li. Global existence and large time behavior of a 2D Keller-Segel system in logarithmic Lebesgue spaces. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 183-195. doi: 10.3934/dcdsb.2018093
[4]

Luca Battaglia. A general existence result for stationary solutions to the Keller-Segel system. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 905-926. doi: 10.3934/dcds.2019038
[5]

Hai-Yang Jin, Zhi-An Wang. Global stabilization of the full attraction-repulsion Keller-Segel system. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3509-3527. doi: 10.3934/dcds.2020027
[6]

Hao Yu, Wei Wang, Sining Zheng. Global boundedness of solutions to a Keller-Segel system with nonlinear sensitivity. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1317-1327. doi: 10.3934/dcdsb.2016.21.1317
[7]

Norikazu Saito. Error analysis of a conservative finite-element approximation for the Keller-Segel system of chemotaxis. Communications on Pure & Applied Analysis, 2012, 11 (1) : 339-364. doi: 10.3934/cpaa.2012.11.339
[8]

Zhichun Zhai. Well-posedness for two types of generalized Keller-Segel system of chemotaxis in critical Besov spaces. Communications on Pure & Applied Analysis, 2011, 10 (1) : 287-308. doi: 10.3934/cpaa.2011.10.287
[9]

Sachiko Ishida, Tomomi Yokota. Remarks on the global existence of weak solutions to quasilinear degenerate Keller-Segel systems. Conference Publications, 2013, 2013 (special) : 345-354. doi: 10.3934/proc.2013.2013.345
[10]

Wenbin Lv, Qingyuan Wang. Global existence for a class of Keller-Segel models with signal-dependent motility and general logistic term. Evolution Equations & Control Theory, 2021, 10 (1) : 25-36. doi: 10.3934/eect.2020040
[11]

Yajing Zhang, Xinfu Chen, Jianghao Hao, Xin Lai, Cong Qin. Dynamics of spike in a Keller-Segel's minimal chemotaxis model. Discrete & Continuous Dynamical Systems, 2017, 37 (2) : 1109-1127. doi: 10.3934/dcds.2017046
[12]

Tobias Black. Global generalized solutions to a parabolic-elliptic Keller-Segel system with singular sensitivity. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 119-137. doi: 10.3934/dcdss.2020007
[13]

Xinru Cao. Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 1891-1904. doi: 10.3934/dcds.2015.35.1891
[14]

Piotr Biler, Ignacio Guerra, Grzegorz Karch. Large global-in-time solutions of the parabolic-parabolic Keller-Segel system on the plane. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2117-2126. doi: 10.3934/cpaa.2015.14.2117
[15]

Qi Wang, Lu Zhang, Jingyue Yang, Jia Hu. Global existence and steady states of a two competing species Keller--Segel chemotaxis model. Kinetic & Related Models, 2015, 8 (4) : 777-807. doi: 10.3934/krm.2015.8.777
[16]

Ping Liu, Junping Shi, Zhi-An Wang. Pattern formation of the attraction-repulsion Keller-Segel system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2597-2625. doi: 10.3934/dcdsb.2013.18.2597
[17]

Jinhuan Wang, Li Chen, Liang Hong. Parabolic elliptic type Keller-Segel system on the whole space case. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 1061-1084. doi: 10.3934/dcds.2016.36.1061
[18]

Hongyun Peng, Zhi-An Wang, Kun Zhao, Changjiang Zhu. Boundary layers and stabilization of the singular Keller-Segel system. Kinetic & Related Models, 2018, 11 (5) : 1085-1123. doi: 10.3934/krm.2018042
[19]

Hai-Yang Jin, Zhi-An Wang. The Keller-Segel system with logistic growth and signal-dependent motility. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3023-3041. doi: 10.3934/dcdsb.2020218
[20]

Hao Yu, Wei Wang, Sining Zheng. Boundedness of solutions to a fully parabolic Keller-Segel system with nonlinear sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1635-1644. doi: 10.3934/dcdsb.2017078

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (171)
  • HTML views (85)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences