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
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show all references

References:
[1]

Math. Z., 202 (1989), 219-250. doi: 10.1007/BF01215256.  Google Scholar

[2]

Springer, New York, 2011.  Google Scholar

[3]

J. Math. Anal. Appl., 326 (2007), 1410-1426. doi: 10.1016/j.jmaa.2006.03.080.  Google Scholar

[4]

Ann. Inst. H. Poincaré Anal. Non Lineaire, 27 (2010), 437-446. doi: 10.1016/j.anihpc.2009.11.016.  Google Scholar

[5]

J. Differential Equations, 252 (2012), 5832-5851. doi: 10.1016/j.jde.2012.01.045.  Google Scholar

[6]

Nonlinearity, 21 (2008), 1057-1076. doi: 10.1088/0951-7715/21/5/009.  Google Scholar

[7]

J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.  Google Scholar

[8]

J. Math. Biol., 44 (2002), 463-478. doi: 10.1007/s002850100134.  Google Scholar

[9]

Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.  Google Scholar

[10]

Jahresber. Deutsch. Math.-Verein., 106 (2004), 51-69.  Google Scholar

[11]

J. Differential Equations, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[12]

J. Differential Equations, 252 (2012), 1421-1440. doi: 10.1016/j.jde.2011.02.012.  Google Scholar

[13]

J. Differential Equations, 252 (2012), 2469-2491. 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]

Math. Methods Appl. Sci., 36 (2013), 745-760. doi: 10.1002/mma.2622.  Google Scholar

[16]

J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[17]

IMS Workshop on Reaction-Diffusion Systems (Shatin, 1999). Methods Appl. Anal., 8 (2001), 349-367.  Google Scholar

[18]

Abstr. Appl. Anal., 2006 (2006), 1-21. doi: 10.1155/AAA/2006/23061.  Google Scholar

[19]

Differential Integral Equations, 19 (2006), 841-876.  Google Scholar

[20]

J. Differential Equations, 227 (2006), 333-364. doi: 10.1016/j.jde.2006.03.003.  Google Scholar

[21]

J. Differential Equations, 250 (2011), 3047-3087. doi: 10.1016/j.jde.2011.01.016.  Google Scholar

[22]

J. Differential Equations, 252 (2012), 692-715. doi: 10.1016/j.jde.2011.08.019.  Google Scholar

[23]

Math. Methods Appl. Sci., 33 (2010), 12-24. doi: 10.1002/mma.1146.  Google Scholar

[24]

Journal de Mathématiques Pures et Appliquées, 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020.  Google Scholar

[25]

Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 431-444. doi: 10.1017/S0308210500004649.  Google Scholar

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