Asymptotic profile of solutions to a certain chemotaxis system

Rafał Celiński; Andrzej Raczyński

doi:10.3934/cpaa.2020041

Article Contents

2020, Volume 19, Issue 2: 911-922. Doi: 10.3934/cpaa.2020041

This issue Previous Article Dynamics in a diffusive predator-prey system with stage structure and strong allee effect Next Article Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity

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
^* Corresponding author

Received: November 30, 2018

Revised: May 31, 2019

Published: October 2019

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 92C17, 35B40; Secondary: 35K55, 35Q92.

Citation:

Full Text(HTML)

Related Papers

Cited by

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.