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Emergence of large densities and simultaneous blow-up in a two-species chemotaxis system with competitive kinetics
School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, China |
$ \begin{eqnarray*} \left\{\begin{array}{lll} u_t = \Delta u-\chi_1\nabla\cdot(u\nabla w)+\mu_1u(1-u-a_1v),\ \ \ &x\in \Omega,\ t>0,\\ v_t = \Delta v-\chi_2\nabla\cdot(v\nabla w)+\mu_2v(1-v-a_2u),\ \ &x\in \Omega,\ t>0,\\ w_t = \Delta w-w+u+v,\ \ &x\in \Omega,\ t>0 \end{array}\right. \end{eqnarray*} $ |
$ \Omega\subset\mathbb{R}^n $ |
$ n\geq3 $ |
$ \chi_i, \mu_i, a_i>0 $ |
$ i = 1, 2 $ |
References:
[1] |
X. Bai and M. Winkler,
Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.
doi: 10.1512/iumj.2016.65.5776. |
[2] |
N. Bellomo, A. Bellouquid, Y. Tao and 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.
doi: 10.1142/S021820251550044X. |
[3] |
P. Biler, E. E. Espejo and I. Guerra,
Blowup in higher dimensional two species chemotactic systems, Commun. Pure Appl. Anal., 12 (2013), 89-98.
doi: 10.3934/cpaa.2013.12.89. |
[4] |
P. Biler and I. Guerra,
Blowup and self-similar solutions for two-component drift-diffusion systems, Nonlinear Anal., 75 (2012), 5186-5193.
doi: 10.1016/j.na.2012.04.035. |
[5] |
T. Black, J. Lankeit and M. Mizukami,
On the weakly competitive case in a two-species chemotaxis model, IMA J. Appl. Math., 81 (2016), 860-876.
doi: 10.1093/imamat/hxw036. |
[6] |
C. Conca, E. Espejo and K. Vilches,
Remarks on the blowup and global existence for a two species chemotactic Keller-Segel system in $\Bbb R^2$, European J. Appl. Math., 22 (2011), 553-580.
doi: 10.1017/S0956792511000258. |
[7] |
E. E. Espejo Arenas, A. Stevens and J. J. L. Velázquez,
Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis (Munich), 29 (2009), 317-338.
doi: 10.1524/anly.2009.1029. |
[8] |
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. |
[9] |
D. Horstmann,
Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci., 21 (2011), 231-270.
doi: 10.1007/s00332-010-9082-x. |
[10] |
D. Horstmann and G. Wang,
Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.
doi: 10.1017/S0956792501004363. |
[11] |
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. |
[12] |
J. Lankeit,
Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191.
doi: 10.1016/j.jde.2014.10.016. |
[13] |
Y. Li and J. Lankeit,
Boundedness in a chemotaxis-haptotaxis model with nonlinear diffusion, Nonlinearity, 29 (2016), 1564-1595.
doi: 10.1088/0951-7715/29/5/1564. |
[14] |
Y. Li and Y. Li,
Finite-time blow-up in higher dimensional fully-parabolic chemotaxis system for two species, Nonlinear Anal., 109 (2014), 72-84.
doi: 10.1016/j.na.2014.05.021. |
[15] |
K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura,
Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., 51 (2002), 119-144.
doi: 10.1016/S0362-546X(01)00815-X. |
[16] |
I. G. Pearce, M. A. J. Chaplain, P. G. Schofield, A. R. A. Anderson and S. F. Hubbard,
Chemotaxis-induced spatio-temporal heterogeneity in multi-species host-parasitoid systems, J. Math. Biol., 55 (2007), 365-388.
doi: 10.1007/s00285-007-0088-4. |
[17] |
C. Stinner, J. I. Tello and M. Winkler,
Competitive exclusion in a two-species chemotaxis model, J. Math. Biol., 68 (2014), 1607-1626.
doi: 10.1007/s00285-013-0681-7. |
[18] |
C. Stinner, C. Surulescu and M. Winkler,
Global weak solutions in a pde-ode system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007.
doi: 10.1137/13094058X. |
[19] |
X. Tang and Y. Tao,
Analysis of a chemotaxis model for multi-species host-parasitoid interactions, Appl. Math. Sci., 2 (2008), 1239-1252.
|
[20] |
Y. Tao and M. Winkler,
Boundedness in a quasilinear parabolic-parabolic keller-segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[21] |
J. I. Tello and M. Winkler,
Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425.
doi: 10.1088/0951-7715/25/5/1413. |
[22] |
J. I. Tello and M. Winkler,
A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[23] |
M. Winkler,
Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[24] |
M. Winkler,
Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.
doi: 10.1080/03605300903473426. |
[25] |
M. Winkler,
Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.
doi: 10.1016/j.jmaa.2011.05.057. |
[26] |
M. Winkler,
Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system, J. Math. Pures Appl. (9), 100 (2013), 748-767.
doi: 10.1016/j.matpur.2013.01.020. |
[27] |
M. Winkler,
Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2777-2793.
doi: 10.3934/dcdsb.2017135. |
[28] |
M. Winkler, Finite-time blow-up in low-dimensional keller-segel systems with logistic-type superlinear degradation, Z. Angel. Math. Phy., 69 (2018), Art. 69, 40 pp.
doi: 10.1007/s00033-018-0935-8. |
[29] |
G. Wolansky,
Multi-components chemotactic system in the absence of conflicts, European J. Appl. Math., 13 (2002), 641-661.
doi: 10.1017/S0956792501004843. |
[30] |
Q. Zhang and Y. Li,
Global existence and asymptotic properties of the solution to a two-species chemotaxis system, J. Math. Anal. Appl., 418 (2014), 47-63.
doi: 10.1016/j.jmaa.2014.03.084. |
[31] |
Q. Zhang and Y. Li,
Boundedness in a quasilinear fully parabolic keller–segel system with logistic source, Z. Angew. Math. Phy., 66 (2015), 2473-2484.
doi: 10.1007/s00033-015-0532-z. |
show all references
References:
[1] |
X. Bai and M. Winkler,
Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.
doi: 10.1512/iumj.2016.65.5776. |
[2] |
N. Bellomo, A. Bellouquid, Y. Tao and 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.
doi: 10.1142/S021820251550044X. |
[3] |
P. Biler, E. E. Espejo and I. Guerra,
Blowup in higher dimensional two species chemotactic systems, Commun. Pure Appl. Anal., 12 (2013), 89-98.
doi: 10.3934/cpaa.2013.12.89. |
[4] |
P. Biler and I. Guerra,
Blowup and self-similar solutions for two-component drift-diffusion systems, Nonlinear Anal., 75 (2012), 5186-5193.
doi: 10.1016/j.na.2012.04.035. |
[5] |
T. Black, J. Lankeit and M. Mizukami,
On the weakly competitive case in a two-species chemotaxis model, IMA J. Appl. Math., 81 (2016), 860-876.
doi: 10.1093/imamat/hxw036. |
[6] |
C. Conca, E. Espejo and K. Vilches,
Remarks on the blowup and global existence for a two species chemotactic Keller-Segel system in $\Bbb R^2$, European J. Appl. Math., 22 (2011), 553-580.
doi: 10.1017/S0956792511000258. |
[7] |
E. E. Espejo Arenas, A. Stevens and J. J. L. Velázquez,
Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis (Munich), 29 (2009), 317-338.
doi: 10.1524/anly.2009.1029. |
[8] |
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. |
[9] |
D. Horstmann,
Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci., 21 (2011), 231-270.
doi: 10.1007/s00332-010-9082-x. |
[10] |
D. Horstmann and G. Wang,
Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.
doi: 10.1017/S0956792501004363. |
[11] |
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. |
[12] |
J. Lankeit,
Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191.
doi: 10.1016/j.jde.2014.10.016. |
[13] |
Y. Li and J. Lankeit,
Boundedness in a chemotaxis-haptotaxis model with nonlinear diffusion, Nonlinearity, 29 (2016), 1564-1595.
doi: 10.1088/0951-7715/29/5/1564. |
[14] |
Y. Li and Y. Li,
Finite-time blow-up in higher dimensional fully-parabolic chemotaxis system for two species, Nonlinear Anal., 109 (2014), 72-84.
doi: 10.1016/j.na.2014.05.021. |
[15] |
K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura,
Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., 51 (2002), 119-144.
doi: 10.1016/S0362-546X(01)00815-X. |
[16] |
I. G. Pearce, M. A. J. Chaplain, P. G. Schofield, A. R. A. Anderson and S. F. Hubbard,
Chemotaxis-induced spatio-temporal heterogeneity in multi-species host-parasitoid systems, J. Math. Biol., 55 (2007), 365-388.
doi: 10.1007/s00285-007-0088-4. |
[17] |
C. Stinner, J. I. Tello and M. Winkler,
Competitive exclusion in a two-species chemotaxis model, J. Math. Biol., 68 (2014), 1607-1626.
doi: 10.1007/s00285-013-0681-7. |
[18] |
C. Stinner, C. Surulescu and M. Winkler,
Global weak solutions in a pde-ode system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007.
doi: 10.1137/13094058X. |
[19] |
X. Tang and Y. Tao,
Analysis of a chemotaxis model for multi-species host-parasitoid interactions, Appl. Math. Sci., 2 (2008), 1239-1252.
|
[20] |
Y. Tao and M. Winkler,
Boundedness in a quasilinear parabolic-parabolic keller-segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[21] |
J. I. Tello and M. Winkler,
Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425.
doi: 10.1088/0951-7715/25/5/1413. |
[22] |
J. I. Tello and M. Winkler,
A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[23] |
M. Winkler,
Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[24] |
M. Winkler,
Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.
doi: 10.1080/03605300903473426. |
[25] |
M. Winkler,
Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.
doi: 10.1016/j.jmaa.2011.05.057. |
[26] |
M. Winkler,
Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system, J. Math. Pures Appl. (9), 100 (2013), 748-767.
doi: 10.1016/j.matpur.2013.01.020. |
[27] |
M. Winkler,
Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2777-2793.
doi: 10.3934/dcdsb.2017135. |
[28] |
M. Winkler, Finite-time blow-up in low-dimensional keller-segel systems with logistic-type superlinear degradation, Z. Angel. Math. Phy., 69 (2018), Art. 69, 40 pp.
doi: 10.1007/s00033-018-0935-8. |
[29] |
G. Wolansky,
Multi-components chemotactic system in the absence of conflicts, European J. Appl. Math., 13 (2002), 641-661.
doi: 10.1017/S0956792501004843. |
[30] |
Q. Zhang and Y. Li,
Global existence and asymptotic properties of the solution to a two-species chemotaxis system, J. Math. Anal. Appl., 418 (2014), 47-63.
doi: 10.1016/j.jmaa.2014.03.084. |
[31] |
Q. Zhang and Y. Li,
Boundedness in a quasilinear fully parabolic keller–segel system with logistic source, Z. Angew. Math. Phy., 66 (2015), 2473-2484.
doi: 10.1007/s00033-015-0532-z. |
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