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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

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 & 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.  Google Scholar

[2]

P. Biler, Existence and nonexistence of solutions for a model of gravitational interaction of particles III, Coll. Math., 68 (1995), 229-239.  Google Scholar

[3]

P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743.  Google Scholar

[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.  Google Scholar

[5]

P. Biler, T. Cieślak, G. Karch and J. Zienkiewicz, Local criteria for blowup of solutions in two-dimensional chemotaxis models,, , ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

P. Biler, The Cauchy problem and self-similar solutions for a nonlinear parabolic equation, Studia Math., 114 (1995), 181-205.  Google Scholar

[2]

P. Biler, Existence and nonexistence of solutions for a model of gravitational interaction of particles III, Coll. Math., 68 (1995), 229-239.  Google Scholar

[3]

P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743.  Google Scholar

[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.  Google Scholar

[5]

P. Biler, T. Cieślak, G. Karch and J. Zienkiewicz, Local criteria for blowup of solutions in two-dimensional chemotaxis models,, , ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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