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Pattern formation of the attraction-repulsion Keller-Segel system
Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation
1. | College of Liberal Arts and Sciences, Tokyo Medical and Dental University, 2-8-30 Kohnodai, Ichikawa, Chiba 272-0827, Japan |
2. | Department of Mathematical Sciences, School of Science and Technology, Kwansei Gakuin University, 2-1 Gakuen, Sanda 669-1337, Japan |
References:
[1] |
M. Aida, T. Tsujikawa, M. Efendiev, A. Yagi and M. Mimura, Lower estimate of the attractor dimension for a chemotaxis growth system, J. London Math. Soc. (2), 74 (2006), 453-474.
doi: 10.1112/S0024610706023015. |
[2] |
S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), 217-237.
doi: 10.1016/0025-5564(81)90055-9. |
[3] |
R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 5: Evolution Problems I, With the collaboration of Michel Artola, Michel Cessenat and Hélène Lanchon. Translated from the French by Alan Craig. Springer-Verlag, Berlin, 1992.
doi: 10.1007/978-3-642-58090-1. |
[4] |
A. Eden, C. Foias, B. Nicolaenko and R. Temam, Ensembles inertiels pour des équations d'évolution dissipatives, (French) [Inertial sets for dissipative evolution equations], C. R. Acad. Sci. Paris, 310 (1990), 559-562. |
[5] |
A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, Research in Applied Mathematics, vol. 37, John-Wiley and Sons, Chichester, 1994. |
[6] |
M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scoula Norm. Sup. Pisa Cl. Sci. IV, 24 (1997), 633-683. |
[7] |
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. |
[8] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. |
[9] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences II, Jahresber. Deutsch. Math.-Verein., 106 (2004), 51-69. |
[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] |
W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.
doi: 10.1090/S0002-9947-1992-1046835-6. |
[12] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[13] |
N. Kurata, K. Kuto, K. Osaki, T. Tsujikawa and T. Sakurai, Bifurcation phenomena of pattern solution to Mimura-Tsujikawa model in one dimension, GAKUTO Internat. Ser. Math. Sci. Appl., 29 (2008), 265-278. |
[14] |
K. Kuto, K. Osaki, T. Sakurai and T. Tsujikawa, Spatial pattern formation in a chemotaxis-diffusion-growth model, Physica D, 241 (2012), 1629-1639.
doi: 10.1016/j.physd.2012.06.009. |
[15] |
M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth, Physica A, 230 (1996), 499-543.
doi: 10.1016/0378-4371(96)00051-9. |
[16] |
J. D. Murray, Mathematical Biology, II: Spatial Models and Biomedical Applications, 3rd edition, Springer-Verlag, New York, 2003. |
[17] |
E. Nakaguchi and K. Osaki, Global existence of solutions to a parabolic-parabolic system for chemotaxis with weak degradation, Nonlinear Anal., 74 (2011), 286-297.
doi: 10.1016/j.na.2010.08.044. |
[18] |
T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601. |
[19] |
T. Nagai, Blow up of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domain, J. Inequal. Appl., 6 (2001), 37-55.
doi: 10.1155/S1025583401000042. |
[20] |
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. |
[21] |
K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. |
[22] |
K. Osaki and A. Yagi, Global existence for a chemotaxis-growth system in $\mathbb{R}^2$, Adv. Math. Sci. Appl., 12 (2002), 587-606. |
[23] |
K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica D: Nonlinear Phenomena, 240 (2011), 363-375.
doi: 10.1016/j.physd.2010.09.011. |
[24] |
T. Suzuki, Free Energy and Self-Interacting Particles, Progress in Nonlinear Differential Equations and their Applications, 62. Birkhäuser Boston, Inc., Boston, MA, 2005.
doi: 10.1007/0-8176-4436-9. |
[25] |
J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[26] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. |
[27] |
M. J. Tindall, P. K. Maini, S. L. Porter and J. P. Armitage, Overview of mathematical approaches used to model bacterial chemotaxis II: Bacterial populations, Bull. Math. Biol., 70 (2008), 1570-1607.
doi: 10.1007/s11538-008-9322-5. |
[28] |
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, 2nd revised and enlarged edition, Johann Ambrosius Barth Verlag, Heidelberg/Leipzig, 1995. |
[29] |
M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729.
doi: 10.1016/j.jmaa.2008.07.071. |
[30] |
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. |
[31] |
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. |
[32] |
A. Yagi, Abstract Parabolic Evolution Equations and Their Applications, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-642-04631-5. |
show all references
References:
[1] |
M. Aida, T. Tsujikawa, M. Efendiev, A. Yagi and M. Mimura, Lower estimate of the attractor dimension for a chemotaxis growth system, J. London Math. Soc. (2), 74 (2006), 453-474.
doi: 10.1112/S0024610706023015. |
[2] |
S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), 217-237.
doi: 10.1016/0025-5564(81)90055-9. |
[3] |
R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 5: Evolution Problems I, With the collaboration of Michel Artola, Michel Cessenat and Hélène Lanchon. Translated from the French by Alan Craig. Springer-Verlag, Berlin, 1992.
doi: 10.1007/978-3-642-58090-1. |
[4] |
A. Eden, C. Foias, B. Nicolaenko and R. Temam, Ensembles inertiels pour des équations d'évolution dissipatives, (French) [Inertial sets for dissipative evolution equations], C. R. Acad. Sci. Paris, 310 (1990), 559-562. |
[5] |
A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, Research in Applied Mathematics, vol. 37, John-Wiley and Sons, Chichester, 1994. |
[6] |
M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scoula Norm. Sup. Pisa Cl. Sci. IV, 24 (1997), 633-683. |
[7] |
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. |
[8] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. |
[9] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences II, Jahresber. Deutsch. Math.-Verein., 106 (2004), 51-69. |
[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] |
W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.
doi: 10.1090/S0002-9947-1992-1046835-6. |
[12] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[13] |
N. Kurata, K. Kuto, K. Osaki, T. Tsujikawa and T. Sakurai, Bifurcation phenomena of pattern solution to Mimura-Tsujikawa model in one dimension, GAKUTO Internat. Ser. Math. Sci. Appl., 29 (2008), 265-278. |
[14] |
K. Kuto, K. Osaki, T. Sakurai and T. Tsujikawa, Spatial pattern formation in a chemotaxis-diffusion-growth model, Physica D, 241 (2012), 1629-1639.
doi: 10.1016/j.physd.2012.06.009. |
[15] |
M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth, Physica A, 230 (1996), 499-543.
doi: 10.1016/0378-4371(96)00051-9. |
[16] |
J. D. Murray, Mathematical Biology, II: Spatial Models and Biomedical Applications, 3rd edition, Springer-Verlag, New York, 2003. |
[17] |
E. Nakaguchi and K. Osaki, Global existence of solutions to a parabolic-parabolic system for chemotaxis with weak degradation, Nonlinear Anal., 74 (2011), 286-297.
doi: 10.1016/j.na.2010.08.044. |
[18] |
T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601. |
[19] |
T. Nagai, Blow up of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domain, J. Inequal. Appl., 6 (2001), 37-55.
doi: 10.1155/S1025583401000042. |
[20] |
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. |
[21] |
K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. |
[22] |
K. Osaki and A. Yagi, Global existence for a chemotaxis-growth system in $\mathbb{R}^2$, Adv. Math. Sci. Appl., 12 (2002), 587-606. |
[23] |
K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica D: Nonlinear Phenomena, 240 (2011), 363-375.
doi: 10.1016/j.physd.2010.09.011. |
[24] |
T. Suzuki, Free Energy and Self-Interacting Particles, Progress in Nonlinear Differential Equations and their Applications, 62. Birkhäuser Boston, Inc., Boston, MA, 2005.
doi: 10.1007/0-8176-4436-9. |
[25] |
J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[26] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. |
[27] |
M. J. Tindall, P. K. Maini, S. L. Porter and J. P. Armitage, Overview of mathematical approaches used to model bacterial chemotaxis II: Bacterial populations, Bull. Math. Biol., 70 (2008), 1570-1607.
doi: 10.1007/s11538-008-9322-5. |
[28] |
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, 2nd revised and enlarged edition, Johann Ambrosius Barth Verlag, Heidelberg/Leipzig, 1995. |
[29] |
M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729.
doi: 10.1016/j.jmaa.2008.07.071. |
[30] |
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. |
[31] |
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. |
[32] |
A. Yagi, Abstract Parabolic Evolution Equations and Their Applications, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-642-04631-5. |
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