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Dynamics of hyperbolic meromorphic functions
Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source
1. | College of Sciences, Chongqing University of Posts and Telecommunications, Chongqing 400065, China, China |
2. | College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China |
References:
[1] |
N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.
doi: 10.1080/03605307908820113. |
[2] |
K. Baghaei and M. Hesaaraki, Global existence and boundedness of classical solutions for a chemotaxis model with logistic source, C. R. Acad. Sci. Paris, Ser. I, 351 (2013), 585-591.
doi: 10.1016/j.crma.2013.07.027. |
[3] |
J. Burczak, T. Cieślak and C. Morales-Rodrigo, Global existence vs. blow-up in a fully parabolic quasilinear 1D Keller-Segel system, Nonlinear Anal., 75 (2012), 5215-5228.
doi: 10.1016/j.na.2012.04.038. |
[4] |
V. Calvez and J. A. Carrillo, Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math. Pures Appl., 86 (2006), 155-175.
doi: 10.1016/j.matpur.2006.04.002. |
[5] |
X. Cao, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with logistic source, J. Math. Anal. Appl., 412 (2014), 181-188.
doi: 10.1016/j.jmaa.2013.10.061. |
[6] |
X. Cao and S. Zheng, Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, Math. Methods Appl. Sci., 37 (2014), 2326-2330.
doi: 10.1002/mma.2992. |
[7] |
T. Cieślak and P. Laurençot, Finite-time blow-up for one-dimensional quasilinear parabolic-parabolic chemotaxis system, Ann. Inst. H.Poincaré Anal. Non Linéarire, 27 (2010), 437-446.
doi: 10.1016/j.anihpc.2009.11.016. |
[8] |
T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851.
doi: 10.1016/j.jde.2012.01.045. |
[9] |
T. Cieślak and C. Stinner, Finite-time blowup in a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 2, Acta Appl. Math., 129 (2014), 135-146.
doi: 10.1007/s10440-013-9832-5. |
[10] |
T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076.
doi: 10.1088/0951-7715/21/5/009. |
[11] |
A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969. |
[12] |
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. |
[13] |
D. Horstmann, On the existence of radially symmetric blow-up solutions for the Keller-Segel model, J. Math. Biol., 44 (2002), 463-478.
doi: 10.1007/s002850100134. |
[14] |
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. |
[15] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I, Jahresber. Deutsch. Math. -Verein., 105 (2003), 103-165. |
[16] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences II, Jahresber. Deutsch. Math. -Verein., 106 (2004), 51-69. |
[17] |
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. |
[18] |
D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[19] |
S. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domain, J. Differential Equations, 256 (2014), 2993-3010.
doi: 10.1016/j.jde.2014.01.028. |
[20] |
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. |
[21] |
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. |
[22] |
J. Lankeit, Chemotaxis can prevent thresholds on population density,, , ().
|
[23] |
C. Mu, L. Wang, P. Zheng and Q. Zhang, Global existence and boundedness of classical solutions to a parabolic-parabolic chemotaxis system, Nonlinear Anal. Real World Appl., 14 (2013), 1634-1642.
doi: 10.1016/j.nonrwa.2012.10.022. |
[24] |
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. |
[25] |
L. Nirenberg, An extended interpolation inequality, Ann. Sc. Norm. Super. Pisa Cl. Sci., (3) 20 (1966), 733-737. |
[26] |
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. |
[27] |
K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543. |
[28] |
C. S. Patlak, Random walk with persistence and external bias, Bull. Math. BiophyS., 15 (1953), 311-338.
doi: 10.1007/BF02476407. |
[29] |
Y. Tao and Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.
doi: 10.1142/S0218202512500443. |
[30] |
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. |
[31] |
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. |
[32] |
J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[33] |
R. Temam, Infinite-dimensional Dynamical Dystems in Mechanics and Physics, 2nd ed., Appl. Math. Sci., vol. 68, Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[34] |
L. Wang, Y. Li and C. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 789-802.
doi: 10.3934/dcds.2014.34.789. |
[35] |
L. Wang, C. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872.
doi: 10.1016/j.jde.2013.12.007. |
[36] |
Z. A. Wang, On chemotaxis models with cell population interactions, Math. Model. Nat. Phenom., 5 (2010), 173-190.
doi: 10.1051/mmnp/20105311. |
[37] |
Z. A. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos, 17 (2007), 037108, 13pp.
doi: 10.1063/1.2766864. |
[38] |
Z. A. Wang, M. Winkler and D. Wrzosek, Singularity formation in chemotaxis systems with volume-filling effect, Nonlinearity, 24 (2011), 3279-3297.
doi: 10.1088/0951-7715/24/12/001. |
[39] |
Z. A. Wang, M. Winkler and D. Wrzosek, Global regularity vs. infinite-time singularity formation in a chemotaxis model with volume-filling effect and degenerate diffusion, SIAM J. Math. Anal., 44 (2012), 3502-3525.
doi: 10.1137/110853972. |
[40] |
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. |
[41] |
M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.
doi: 10.1016/j.matpur.2013.01.020. |
[42] |
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. |
[43] |
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. |
[44] |
M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr., 283 (2010), 1664-1673.
doi: 10.1002/mana.200810838. |
[45] |
M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.
doi: 10.1002/mma.1146. |
[46] |
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. |
[47] |
M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855.
doi: 10.1007/s00332-014-9205-x. |
[48] |
M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064.
doi: 10.1016/j.na.2009.07.045. |
[49] |
D. Wrzosek, Model of chemotaxis with threshold density and singular diffusion, Nonlinear Anal., 73 (2010), 338-349.
doi: 10.1016/j.na.2010.02.047. |
[50] |
D. Wrzosek, Volume filling effect in modelling chemotaxis, Math. Model. Nat. Phenom., 5 (2010), 123-147.
doi: 10.1051/mmnp/20105106. |
[51] |
P. Zheng, C. Mu and L. Wang, Finite-time blow-up and global boundedness for a quasilinear parabolic-elliptic chemotaxis system with logistic source,, preprint., ().
|
show all references
References:
[1] |
N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.
doi: 10.1080/03605307908820113. |
[2] |
K. Baghaei and M. Hesaaraki, Global existence and boundedness of classical solutions for a chemotaxis model with logistic source, C. R. Acad. Sci. Paris, Ser. I, 351 (2013), 585-591.
doi: 10.1016/j.crma.2013.07.027. |
[3] |
J. Burczak, T. Cieślak and C. Morales-Rodrigo, Global existence vs. blow-up in a fully parabolic quasilinear 1D Keller-Segel system, Nonlinear Anal., 75 (2012), 5215-5228.
doi: 10.1016/j.na.2012.04.038. |
[4] |
V. Calvez and J. A. Carrillo, Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math. Pures Appl., 86 (2006), 155-175.
doi: 10.1016/j.matpur.2006.04.002. |
[5] |
X. Cao, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with logistic source, J. Math. Anal. Appl., 412 (2014), 181-188.
doi: 10.1016/j.jmaa.2013.10.061. |
[6] |
X. Cao and S. Zheng, Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, Math. Methods Appl. Sci., 37 (2014), 2326-2330.
doi: 10.1002/mma.2992. |
[7] |
T. Cieślak and P. Laurençot, Finite-time blow-up for one-dimensional quasilinear parabolic-parabolic chemotaxis system, Ann. Inst. H.Poincaré Anal. Non Linéarire, 27 (2010), 437-446.
doi: 10.1016/j.anihpc.2009.11.016. |
[8] |
T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851.
doi: 10.1016/j.jde.2012.01.045. |
[9] |
T. Cieślak and C. Stinner, Finite-time blowup in a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 2, Acta Appl. Math., 129 (2014), 135-146.
doi: 10.1007/s10440-013-9832-5. |
[10] |
T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076.
doi: 10.1088/0951-7715/21/5/009. |
[11] |
A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969. |
[12] |
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. |
[13] |
D. Horstmann, On the existence of radially symmetric blow-up solutions for the Keller-Segel model, J. Math. Biol., 44 (2002), 463-478.
doi: 10.1007/s002850100134. |
[14] |
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. |
[15] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I, Jahresber. Deutsch. Math. -Verein., 105 (2003), 103-165. |
[16] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences II, Jahresber. Deutsch. Math. -Verein., 106 (2004), 51-69. |
[17] |
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. |
[18] |
D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[19] |
S. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domain, J. Differential Equations, 256 (2014), 2993-3010.
doi: 10.1016/j.jde.2014.01.028. |
[20] |
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. |
[21] |
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. |
[22] |
J. Lankeit, Chemotaxis can prevent thresholds on population density,, , ().
|
[23] |
C. Mu, L. Wang, P. Zheng and Q. Zhang, Global existence and boundedness of classical solutions to a parabolic-parabolic chemotaxis system, Nonlinear Anal. Real World Appl., 14 (2013), 1634-1642.
doi: 10.1016/j.nonrwa.2012.10.022. |
[24] |
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. |
[25] |
L. Nirenberg, An extended interpolation inequality, Ann. Sc. Norm. Super. Pisa Cl. Sci., (3) 20 (1966), 733-737. |
[26] |
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. |
[27] |
K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543. |
[28] |
C. S. Patlak, Random walk with persistence and external bias, Bull. Math. BiophyS., 15 (1953), 311-338.
doi: 10.1007/BF02476407. |
[29] |
Y. Tao and Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.
doi: 10.1142/S0218202512500443. |
[30] |
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. |
[31] |
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. |
[32] |
J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[33] |
R. Temam, Infinite-dimensional Dynamical Dystems in Mechanics and Physics, 2nd ed., Appl. Math. Sci., vol. 68, Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[34] |
L. Wang, Y. Li and C. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 789-802.
doi: 10.3934/dcds.2014.34.789. |
[35] |
L. Wang, C. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872.
doi: 10.1016/j.jde.2013.12.007. |
[36] |
Z. A. Wang, On chemotaxis models with cell population interactions, Math. Model. Nat. Phenom., 5 (2010), 173-190.
doi: 10.1051/mmnp/20105311. |
[37] |
Z. A. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos, 17 (2007), 037108, 13pp.
doi: 10.1063/1.2766864. |
[38] |
Z. A. Wang, M. Winkler and D. Wrzosek, Singularity formation in chemotaxis systems with volume-filling effect, Nonlinearity, 24 (2011), 3279-3297.
doi: 10.1088/0951-7715/24/12/001. |
[39] |
Z. A. Wang, M. Winkler and D. Wrzosek, Global regularity vs. infinite-time singularity formation in a chemotaxis model with volume-filling effect and degenerate diffusion, SIAM J. Math. Anal., 44 (2012), 3502-3525.
doi: 10.1137/110853972. |
[40] |
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. |
[41] |
M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.
doi: 10.1016/j.matpur.2013.01.020. |
[42] |
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. |
[43] |
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. |
[44] |
M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr., 283 (2010), 1664-1673.
doi: 10.1002/mana.200810838. |
[45] |
M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.
doi: 10.1002/mma.1146. |
[46] |
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. |
[47] |
M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855.
doi: 10.1007/s00332-014-9205-x. |
[48] |
M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064.
doi: 10.1016/j.na.2009.07.045. |
[49] |
D. Wrzosek, Model of chemotaxis with threshold density and singular diffusion, Nonlinear Anal., 73 (2010), 338-349.
doi: 10.1016/j.na.2010.02.047. |
[50] |
D. Wrzosek, Volume filling effect in modelling chemotaxis, Math. Model. Nat. Phenom., 5 (2010), 123-147.
doi: 10.1051/mmnp/20105106. |
[51] |
P. Zheng, C. Mu and L. Wang, Finite-time blow-up and global boundedness for a quasilinear parabolic-elliptic chemotaxis system with logistic source,, preprint., ().
|
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