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Dynamics in a diffusive predator-prey system with stage structure and strong allee effect
Asymptotic profile of solutions to a certain chemotaxis system
Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, POLAND |
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.
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. |
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. |
| [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. |
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