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Boundedness in a quasilinear fully parabolic Keller-Segel system via maximal Sobolev regularity
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On a chemotaxis model with competitive terms arising in angiogenesis
Global asymptotic stability in a chemotaxis-growth model for tumor invasion
Department of Mathematics, Tokyo University of Science, Tokyo 162-8601, Japan |
$ \left\{ \begin{array}{l} u_t = \Delta u - \nabla \cdot (u\nabla v) + ru -\mu u^\alpha, \qquad x\in \Omega, \ t>0, \\ \ v_t = \Delta v + wz, \qquad x\in \Omega, \ t>0, \\ \ w_t = -wz, \qquad x\in \Omega, \ t>0, \\ \ z_t = \Delta z - z + u, \qquad x\in \Omega, \ t>0, \end{array} \right. $ |
$ \Omega \subset \mathbb{R}^n $ |
$ n \le 3 $ |
$ r>0 $ |
$ \mu>0 $ |
$ \alpha>1 $ |
$ ru-\mu u^\alpha $ |
References:
[1] |
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. |
[2] |
X. Cao,
Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.
doi: 10.3934/dcds.2015.35.1891. |
[3] |
M. A. J. Chaplain and A. R. A. Anderson, Mathematical modelling of tissue invasion, in Cancer modelling and simulation, Chapman & Hall/CRC Math. Biol. Med. Ser., Chapman & Hall/CRC, Boca Raton, FL, (2003), 269-297. |
[4] |
E. Feireisl, P. Laurençot and H. Petzeltová,
On convergence to equilibria for the Keller-Segel chemotaxis model, J. Differential Equations, 236 (2007), 551-569.
doi: 10.1016/j.jde.2007.02.002. |
[5] |
A. Friedman and J. I. Tello,
Stability of solutions of chemotaxis equations in reinforced random walks, J. Math. Anal. Appl., 272 (2002), 138-163.
doi: 10.1016/S0022-247X(02)00147-6. |
[6] |
K. Fujie, A. Ito and T. Yokota,
Existence and uniqueness of local classical solutions to modified tumor invasion models of Chaplain-Anderson type, Adv. Math. Sci. Appl., 24 (2014), 67-84.
|
[7] |
K. Fujie, A. Ito, M. Winkler and T. Yokota,
Stabilization in a chemotaxis model for tumor invasion, Discrete Contin. Dyn. Syst., 36 (2016), 151-169.
doi: 10.3934/dcds.2016.36.151. |
[8] |
K. Fujie and T. Senba,
Application of an Adams type inequality to a two-chemical substances chemotaxis system, J. Differential Equations, 263 (2017), 88-148.
doi: 10.1016/j.jde.2017.02.031. |
[9] |
M. A. Herrero and J. J. L. Velázquez,
A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633-683.
|
[10] |
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. |
[11] |
D. Horstmann,
From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.
|
[12] |
B. Hu and Y. Tao,
To the exclusion of blow-up in a three-dimensional chemotaxis-growth model with indirect attractant production, Math. Models Methods Appl. Sci., 26 (2016), 2111-2128.
doi: 10.1142/S0218202516400091. |
[13] |
K. Kang, A. Stevens and J. J. L. Velázquez,
Qualitative behavior of a Keller-Segel model with non-diffusive memory, Commun. Partial Differ. Equations, 35 (2010), 245-274.
doi: 10.1080/03605300903473400. |
[14] |
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. |
[15] |
O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, Amer. Math. Soc. Transl., Vol. 23, Providence, RI, 1968. |
[16] |
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. |
[17] |
G. Liţcanu and C. Morales-Rodrigo,
Asymptotic behaviour of global solutions to a model of cell invasion, Math. Mod. Meth. Appl. Sci., 20 (2010), 1721-1758.
doi: 10.1142/S0218202510004775. |
[18] |
A. Marciniak-Czochra and M. Ptashnyk,
Boundedness of solutions of a haptotaxis model, Math. Models Methods Appl. Sci., 20 (2010), 449-476.
doi: 10.1142/S0218202510004301. |
[19] |
C. Morales-Rodrigo,
Local existence and uniqueness of regular solutions in a model of tissue invasion by solid tumours, Math. Comput. Modelling, 47 (2008), 604-613.
doi: 10.1016/j.mcm.2007.02.031. |
[20] |
K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura,
Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., Theory Methods Appl., 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, Funkcialaj Ekvacioj, 44 (2001), 441-469.
|
[22] |
Z. Szymańska, C. Morales-Rodrigo, M. Lachowicz and M. A. J. Chaplain,
Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interactions, Math. Models Methods Appl. Sci., 19 (2009), 257-281.
doi: 10.1142/S0218202509003425. |
[23] |
Y. Tao,
Global existence for a haptotaxis model of cancer invasion with tissue remodeling, Nonlinear Anal. Real World Appl., 12 (2011), 418-435.
doi: 10.1016/j.nonrwa.2010.06.027. |
[24] |
Y. Tao and M. Winkler,
Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production, J. Eur. Math. Soc.(JEMS), 19 (2017), 3641-3678.
doi: 10.4171/JEMS/749. |
[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] |
J. I. Tello and D. Wrzosek,
Predator-prey model with diffusion and indirect prey-taxis, Math. Models Methods Appl. Sci., 26 (2016), 2129-2162.
doi: 10.1142/S0218202516400108. |
[27] |
C. Walker and G. F. Webb,
Global existence of classical solutions for a haptotaxis model, SIAM J. Math. Anal., 38 (2007), 1694-1713.
doi: 10.1137/060655122. |
[28] |
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. |
[29] |
M. Winkler,
Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, Journal de Mathématiques Pures et Appliquées, 100 (2013), 748-767.
doi: 10.1016/j.matpur.2013.01.020. |
[30] |
M. Winkler,
Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077.
doi: 10.1016/j.jde.2014.04.023. |
show all references
References:
[1] |
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. |
[2] |
X. Cao,
Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.
doi: 10.3934/dcds.2015.35.1891. |
[3] |
M. A. J. Chaplain and A. R. A. Anderson, Mathematical modelling of tissue invasion, in Cancer modelling and simulation, Chapman & Hall/CRC Math. Biol. Med. Ser., Chapman & Hall/CRC, Boca Raton, FL, (2003), 269-297. |
[4] |
E. Feireisl, P. Laurençot and H. Petzeltová,
On convergence to equilibria for the Keller-Segel chemotaxis model, J. Differential Equations, 236 (2007), 551-569.
doi: 10.1016/j.jde.2007.02.002. |
[5] |
A. Friedman and J. I. Tello,
Stability of solutions of chemotaxis equations in reinforced random walks, J. Math. Anal. Appl., 272 (2002), 138-163.
doi: 10.1016/S0022-247X(02)00147-6. |
[6] |
K. Fujie, A. Ito and T. Yokota,
Existence and uniqueness of local classical solutions to modified tumor invasion models of Chaplain-Anderson type, Adv. Math. Sci. Appl., 24 (2014), 67-84.
|
[7] |
K. Fujie, A. Ito, M. Winkler and T. Yokota,
Stabilization in a chemotaxis model for tumor invasion, Discrete Contin. Dyn. Syst., 36 (2016), 151-169.
doi: 10.3934/dcds.2016.36.151. |
[8] |
K. Fujie and T. Senba,
Application of an Adams type inequality to a two-chemical substances chemotaxis system, J. Differential Equations, 263 (2017), 88-148.
doi: 10.1016/j.jde.2017.02.031. |
[9] |
M. A. Herrero and J. J. L. Velázquez,
A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633-683.
|
[10] |
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. |
[11] |
D. Horstmann,
From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.
|
[12] |
B. Hu and Y. Tao,
To the exclusion of blow-up in a three-dimensional chemotaxis-growth model with indirect attractant production, Math. Models Methods Appl. Sci., 26 (2016), 2111-2128.
doi: 10.1142/S0218202516400091. |
[13] |
K. Kang, A. Stevens and J. J. L. Velázquez,
Qualitative behavior of a Keller-Segel model with non-diffusive memory, Commun. Partial Differ. Equations, 35 (2010), 245-274.
doi: 10.1080/03605300903473400. |
[14] |
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. |
[15] |
O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, Amer. Math. Soc. Transl., Vol. 23, Providence, RI, 1968. |
[16] |
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. |
[17] |
G. Liţcanu and C. Morales-Rodrigo,
Asymptotic behaviour of global solutions to a model of cell invasion, Math. Mod. Meth. Appl. Sci., 20 (2010), 1721-1758.
doi: 10.1142/S0218202510004775. |
[18] |
A. Marciniak-Czochra and M. Ptashnyk,
Boundedness of solutions of a haptotaxis model, Math. Models Methods Appl. Sci., 20 (2010), 449-476.
doi: 10.1142/S0218202510004301. |
[19] |
C. Morales-Rodrigo,
Local existence and uniqueness of regular solutions in a model of tissue invasion by solid tumours, Math. Comput. Modelling, 47 (2008), 604-613.
doi: 10.1016/j.mcm.2007.02.031. |
[20] |
K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura,
Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., Theory Methods Appl., 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, Funkcialaj Ekvacioj, 44 (2001), 441-469.
|
[22] |
Z. Szymańska, C. Morales-Rodrigo, M. Lachowicz and M. A. J. Chaplain,
Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interactions, Math. Models Methods Appl. Sci., 19 (2009), 257-281.
doi: 10.1142/S0218202509003425. |
[23] |
Y. Tao,
Global existence for a haptotaxis model of cancer invasion with tissue remodeling, Nonlinear Anal. Real World Appl., 12 (2011), 418-435.
doi: 10.1016/j.nonrwa.2010.06.027. |
[24] |
Y. Tao and M. Winkler,
Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production, J. Eur. Math. Soc.(JEMS), 19 (2017), 3641-3678.
doi: 10.4171/JEMS/749. |
[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] |
J. I. Tello and D. Wrzosek,
Predator-prey model with diffusion and indirect prey-taxis, Math. Models Methods Appl. Sci., 26 (2016), 2129-2162.
doi: 10.1142/S0218202516400108. |
[27] |
C. Walker and G. F. Webb,
Global existence of classical solutions for a haptotaxis model, SIAM J. Math. Anal., 38 (2007), 1694-1713.
doi: 10.1137/060655122. |
[28] |
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. |
[29] |
M. Winkler,
Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, Journal de Mathématiques Pures et Appliquées, 100 (2013), 748-767.
doi: 10.1016/j.matpur.2013.01.020. |
[30] |
M. Winkler,
Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077.
doi: 10.1016/j.jde.2014.04.023. |
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