Morrey spaces norms and criteria for blowup in chemotaxis models

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June 2016, 11(2): 239-250. doi: 10.3934/nhm.2016.11.239

Morrey spaces norms and criteria for blowup in chemotaxis models

Piotr Biler ^1, , Grzegorz Karch ^2, and Jacek Zienkiewicz ^2,

1.	Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50--384 Wrocław
2.	Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

Received April 2015 Revised July 2015 Published March 2016

Abstract
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Two-dimensional Keller--Segel models for the chemotaxis with fractional (anomalous) diffusion are considered. Criteria for blowup of solutions in terms of suitable Morrey spaces norms are derived. Similarly, a criterion for blowup of solutions in terms of the radial initial concentrations, related to suitable Morrey spaces norms, is shown for radially symmetric solutions of chemotaxis in several dimensions. Those conditions are, in a sense, complementary to the ones guaranteeing the global-in-time existence of solutions.

Keywords: fractional diffusion., Chemotaxis, blowup of solutions, parabolic-elliptic Keller-Segel model.

Mathematics Subject Classification: Primary: 35Q92; Secondary: 35B44, 35K55, 35K4.

Citation: Piotr Biler, Grzegorz Karch, Jacek Zienkiewicz. Morrey spaces norms and criteria for blowup in chemotaxis models. Networks and Heterogeneous Media, 2016, 11 (2) : 239-250. doi: 10.3934/nhm.2016.11.239

References:

[1]	P. Biler, The Cauchy problem and self-similar solutions for a nonlinear parabolic equation, Studia Math., 114 (1995), 181-205.
[2]	P. Biler, Existence and nonexistence of solutions for a model of gravitational interaction of particles III, Coll. Math., 68 (1995), 229-239.
[3]	P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743.
[4]	P. Biler and G. Karch, Blowup of solutions to generalized Keller-Segel model, J. Evol. Equ., 10 (2010), 247-262. doi: 10.1007/s00028-009-0048-0.
[5]	P. Biler, T. Cieślak, G. Karch and J. Zienkiewicz, Local criteria for blowup of solutions in two-dimensional chemotaxis models, arXiv:1410.7807 v.2.
[6]	P. Biler, G. Karch and J. Zienkiewicz, Optimal criteria for blowup of radial and $N$-symmetric solutions of chemotaxis systems, Nonlinearity, 28 (2015), 4369-4387. doi: 10.1088/0951-7715/28/12/4369.
[7]	P. Biler and G. Wu, Two-dimensional chemotaxis models with fractional diffusion, Math. Methods Appl. Sci., 32 (2009), 112-126. doi: 10.1002/mma.1036.
[8]	P. Biler and J. Zienkiewicz, Existence of solutions for the Keller-Segel model of chemotaxis with measures as initial data, Bull. Pol. Acad. Sci., Mathematics, 63 (2015), 41-51. doi: 10.4064/ba63-1-6.
[9]	G. Karch and K. Suzuki, Blow-up versus global existence of solutions to aggregation equations, Appl. Math. (Warsaw), 38 (2011), 243-258. doi: 10.4064/am38-3-1.
[10]	M. Kurokiba and T. Ogawa, Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type, Differ. Integral Equ., 16 (2003), 427-452.
[11]	P.-G. Lemarié-Rieusset, Small data in an optimal Banach space for the parabolic-parabolic and parabolic-elliptic Keller-Segel equations in the whole space, Adv. Diff. Equ., 18 (2013), 1189-1208.
[12]	T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55. doi: 10.1155/S1025583401000042.

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References:

[1]	P. Biler, The Cauchy problem and self-similar solutions for a nonlinear parabolic equation, Studia Math., 114 (1995), 181-205.
[2]	P. Biler, Existence and nonexistence of solutions for a model of gravitational interaction of particles III, Coll. Math., 68 (1995), 229-239.
[3]	P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743.
[4]	P. Biler and G. Karch, Blowup of solutions to generalized Keller-Segel model, J. Evol. Equ., 10 (2010), 247-262. doi: 10.1007/s00028-009-0048-0.
[5]	P. Biler, T. Cieślak, G. Karch and J. Zienkiewicz, Local criteria for blowup of solutions in two-dimensional chemotaxis models, arXiv:1410.7807 v.2.
[6]	P. Biler, G. Karch and J. Zienkiewicz, Optimal criteria for blowup of radial and $N$-symmetric solutions of chemotaxis systems, Nonlinearity, 28 (2015), 4369-4387. doi: 10.1088/0951-7715/28/12/4369.
[7]	P. Biler and G. Wu, Two-dimensional chemotaxis models with fractional diffusion, Math. Methods Appl. Sci., 32 (2009), 112-126. doi: 10.1002/mma.1036.
[8]	P. Biler and J. Zienkiewicz, Existence of solutions for the Keller-Segel model of chemotaxis with measures as initial data, Bull. Pol. Acad. Sci., Mathematics, 63 (2015), 41-51. doi: 10.4064/ba63-1-6.
[9]	G. Karch and K. Suzuki, Blow-up versus global existence of solutions to aggregation equations, Appl. Math. (Warsaw), 38 (2011), 243-258. doi: 10.4064/am38-3-1.
[10]	M. Kurokiba and T. Ogawa, Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type, Differ. Integral Equ., 16 (2003), 427-452.
[11]	P.-G. Lemarié-Rieusset, Small data in an optimal Banach space for the parabolic-parabolic and parabolic-elliptic Keller-Segel equations in the whole space, Adv. Diff. Equ., 18 (2013), 1189-1208.
[12]	T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55. doi: 10.1155/S1025583401000042.

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