December  2013, 18(10): 2569-2596. doi: 10.3934/dcdsb.2013.18.2569

Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type

1. 

Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601

Received  November 2012 Revised  July 2013 Published  October 2013

This paper gives a blow-up result for the quasilinear degenerate Keller-Segel systems of parabolic-parabolic type. It is known that the system has a global solvability in the case where $q < m + \frac{2}{N}$ ($m$ denotes the intensity of diffusion and $q$ denotes the nonlinearity) without any restriction on the size of initial data, and where $q \geq m + \frac{2}{N}$ and the initial data are ``small''. However, there is no result when $q \geq m + \frac{2}{N}$ and the initial data are ``large''. This paper discusses such case and shows that there exist blow-up energy solutions from initial data having large negative energy.
Citation: Sachiko Ishida, Tomomi Yokota. Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2569-2596. doi: 10.3934/dcdsb.2013.18.2569
References:
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H. Amann, Dynamic theory of quasilinear parabolic systems. III. Global existence,, Math. Z., 202 (1989), 219.  doi: 10.1007/BF01215256.  Google Scholar

[2]

H. Brezis, Functional Analysis, Sobolev spaces and Partial differential equations,, Springer, (2011).   Google Scholar

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T. Cieślak, Quasilinear nonuniformly parabolic system modelling chemotaxis,, J. Math. Anal. Appl., 326 (2007), 1410.  doi: 10.1016/j.jmaa.2006.03.080.  Google Scholar

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T. Cieślak and P. Laurençot, Finite time blow-up for a one-dimensional quasilinear parabolic-parabolic chemotaxis system,, Ann. Inst. H. Poincaré Anal. Non Lineaire, 27 (2010), 437.  doi: 10.1016/j.anihpc.2009.11.016.  Google Scholar

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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.  doi: 10.1016/j.jde.2012.01.045.  Google Scholar

[6]

T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis,, Nonlinearity, 21 (2008), 1057.  doi: 10.1088/0951-7715/21/5/009.  Google Scholar

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T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[8]

D. Horstmann, On the existence of radially symmetric blow-up solutions for the Keller-Segel model,, J. Math. Biol., 44 (2002), 463.  doi: 10.1007/s002850100134.  Google Scholar

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D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I,, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103.   Google Scholar

[10]

D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. II,, Jahresber. Deutsch. Math.-Verein., 106 (2004), 51.   Google Scholar

[11]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, J. Differential Equations, 215 (2005), 52.  doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[12]

S. Ishida and T. Yokota, Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type,, J. Differential Equations, 252 (2012), 1421.  doi: 10.1016/j.jde.2011.02.012.  Google Scholar

[13]

S. Ishida and T. Yokota, Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type with small data,, J. Differential Equations, 252 (2012), 2469.  doi: 10.1016/j.jde.2011.08.047.  Google Scholar

[14]

S. Ishida and T. Yokota, Remarks on the global existence of weak solutions to quasilinear degenerate Keller-Segel systems,, AIMS proceedings, ().   Google Scholar

[15]

S. Ishida, T. Ono and T. Yokota, Possibility of the existence of blow-up solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type,, Math. Methods Appl. Sci., 36 (2013), 745.  doi: 10.1002/mma.2622.  Google Scholar

[16]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[17]

T. Senba and T. Suzuki, Parabolic system of chemotaxis: Blowup in a finite and the infinite time,, IMS Workshop on Reaction-Diffusion Systems (Shatin, 8 (2001), 349.   Google Scholar

[18]

T. Senba and T. Suzuki, A quasi-linear parabolic system of chemotaxis,, Abstr. Appl. Anal., 2006 (2006), 1.  doi: 10.1155/AAA/2006/23061.  Google Scholar

[19]

Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems,, Differential Integral Equations, 19 (2006), 841.   Google Scholar

[20]

Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate Keller-Segel model with a power factor in drift term,, J. Differential Equations, 227 (2006), 333.  doi: 10.1016/j.jde.2006.03.003.  Google Scholar

[21]

Y. Sugiyama and Y. Yahagi, Extinction, decay and blow-up for Keller-Segel systems of fast diffusion type,, J. Differential Equations, 250 (2011), 3047.  doi: 10.1016/j.jde.2011.01.016.  Google Scholar

[22]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity,, J. Differential Equations, 252 (2012), 692.  doi: 10.1016/j.jde.2011.08.019.  Google Scholar

[23]

M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12.  doi: 10.1002/mma.1146.  Google Scholar

[24]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, Journal de Mathématiques Pures et Appliquées, 100 (2013), 748.  doi: 10.1016/j.matpur.2013.01.020.  Google Scholar

[25]

D. Wrzosek, Long-time behaviour of solutions to a chemotaxis model with volume-filling effect,, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 431.  doi: 10.1017/S0308210500004649.  Google Scholar

show all references

References:
[1]

H. Amann, Dynamic theory of quasilinear parabolic systems. III. Global existence,, Math. Z., 202 (1989), 219.  doi: 10.1007/BF01215256.  Google Scholar

[2]

H. Brezis, Functional Analysis, Sobolev spaces and Partial differential equations,, Springer, (2011).   Google Scholar

[3]

T. Cieślak, Quasilinear nonuniformly parabolic system modelling chemotaxis,, J. Math. Anal. Appl., 326 (2007), 1410.  doi: 10.1016/j.jmaa.2006.03.080.  Google Scholar

[4]

T. Cieślak and P. Laurençot, Finite time blow-up for a one-dimensional quasilinear parabolic-parabolic chemotaxis system,, Ann. Inst. H. Poincaré Anal. Non Lineaire, 27 (2010), 437.  doi: 10.1016/j.anihpc.2009.11.016.  Google Scholar

[5]

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.  doi: 10.1016/j.jde.2012.01.045.  Google Scholar

[6]

T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis,, Nonlinearity, 21 (2008), 1057.  doi: 10.1088/0951-7715/21/5/009.  Google Scholar

[7]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[8]

D. Horstmann, On the existence of radially symmetric blow-up solutions for the Keller-Segel model,, J. Math. Biol., 44 (2002), 463.  doi: 10.1007/s002850100134.  Google Scholar

[9]

D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I,, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103.   Google Scholar

[10]

D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. II,, Jahresber. Deutsch. Math.-Verein., 106 (2004), 51.   Google Scholar

[11]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, J. Differential Equations, 215 (2005), 52.  doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[12]

S. Ishida and T. Yokota, Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type,, J. Differential Equations, 252 (2012), 1421.  doi: 10.1016/j.jde.2011.02.012.  Google Scholar

[13]

S. Ishida and T. Yokota, Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type with small data,, J. Differential Equations, 252 (2012), 2469.  doi: 10.1016/j.jde.2011.08.047.  Google Scholar

[14]

S. Ishida and T. Yokota, Remarks on the global existence of weak solutions to quasilinear degenerate Keller-Segel systems,, AIMS proceedings, ().   Google Scholar

[15]

S. Ishida, T. Ono and T. Yokota, Possibility of the existence of blow-up solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type,, Math. Methods Appl. Sci., 36 (2013), 745.  doi: 10.1002/mma.2622.  Google Scholar

[16]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[17]

T. Senba and T. Suzuki, Parabolic system of chemotaxis: Blowup in a finite and the infinite time,, IMS Workshop on Reaction-Diffusion Systems (Shatin, 8 (2001), 349.   Google Scholar

[18]

T. Senba and T. Suzuki, A quasi-linear parabolic system of chemotaxis,, Abstr. Appl. Anal., 2006 (2006), 1.  doi: 10.1155/AAA/2006/23061.  Google Scholar

[19]

Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems,, Differential Integral Equations, 19 (2006), 841.   Google Scholar

[20]

Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate Keller-Segel model with a power factor in drift term,, J. Differential Equations, 227 (2006), 333.  doi: 10.1016/j.jde.2006.03.003.  Google Scholar

[21]

Y. Sugiyama and Y. Yahagi, Extinction, decay and blow-up for Keller-Segel systems of fast diffusion type,, J. Differential Equations, 250 (2011), 3047.  doi: 10.1016/j.jde.2011.01.016.  Google Scholar

[22]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity,, J. Differential Equations, 252 (2012), 692.  doi: 10.1016/j.jde.2011.08.019.  Google Scholar

[23]

M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12.  doi: 10.1002/mma.1146.  Google Scholar

[24]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, Journal de Mathématiques Pures et Appliquées, 100 (2013), 748.  doi: 10.1016/j.matpur.2013.01.020.  Google Scholar

[25]

D. Wrzosek, Long-time behaviour of solutions to a chemotaxis model with volume-filling effect,, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 431.  doi: 10.1017/S0308210500004649.  Google Scholar

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