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April  2017, 37(4): 1841-1856. doi: 10.3934/dcds.2017077

Local criteria for blowup in two-dimensional chemotaxis models

Piotr Biler 1,, , Tomasz Cieślak 2, , Grzegorz Karch 1, and  Jacek Zienkiewicz 1,

1. 

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

Instytut Matematyczny, Polska Akademia Nauk, ul. Śniadeckich 8, 00-956 Warsaw, Poland

* Corresponding author: Piotr Biler

Received  January 2015 Revised  November 2016 Published  December 2016

We consider two-dimensional versions of the Keller-Segel model for the chemotaxis with either classical (Brownian) or fractional (anomalous) diffusion. Criteria for blowup of solutions in terms of suitable Morrey spaces norms are derived. Moreover, the impact of the consumption term on the global-in-time existence of solutions is analyzed for the classical Keller-Segel system.

Keywords: Chemotaxis, blowup of solutions, global existence of solutions, consumption rate.
Mathematics Subject Classification: Primary:35Q92;Secondary:35B44, 35K55.
Citation: Piotr Biler, Tomasz Cieślak, Grzegorz Karch, Jacek Zienkiewicz. Local criteria for blowup in two-dimensional chemotaxis models. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 1841-1856. doi: 10.3934/dcds.2017077
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, I. Guerra and G. Karch, Large global-in-time solutions of the parabolic-parabolic Keller-Segel system on the plane, Comm. Pure Appl. Analysis, 14 (2015), 2117-2126.  doi: 10.3934/cpaa.2015.14.2117.

[3]

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.

[4]

P. Biler, G. Karch and P. Laurençot, Blowup of solutions to adiffusive aggregation model, Nonlinearity, 22 (2009), 1559-1568.  doi: 10.1088/0951-7715/22/7/003.

[5]

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.

[6]

P. Biler and W. A. Woyczyński, Global and exploding solutions of nonlocal quadratic evolution problems, SIAM J. Appl. Math., 59 (1999), 845-869.  doi: 10.1137/S0036139996313447.

[7]

P. Biler and J. Zienkiewicz, Existence of solutions for the Keller-Segel model of chemotaxis with measures as initial data, Bull. Polish Acad. Sci. Mathematics, 63 (2015), 41-51.  doi: 10.4064/ba63-1-6.

[8]

P. Biler, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations, 44 (2006), 32 pp. 

[9]

Y. Giga, T. Miyakawa and H. Osada, Two-dimensional Navier-Stokes flow with measures as initial vorticity, Arch. Rational Mech. Anal., 104 (1988), 223-250.  doi: 10.1007/BF00281355.

[10]

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.

[11]

H. Kozono and Y. Sugiyama, Local existence and finite time blow-up of solutions in the 2-D Keller-Segel system, J. Evol. Equ., 8 (2008), 353-378.  doi: 10.1007/s00028-008-0375-6.

[12]

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. 

[13]

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. Eq., 18 (2013), 1189-1208. 

[14]

D. Li and J. L. Rodrigo, Finite-time singularities of an aggregation equation in $\mathbb{R}^n$ with fractional dissipation, Comm. Math. Phys., 287 (2009), 687-703.  doi: 10.1007/s00220-008-0669-0.

[15]

D. Li and J. L. Rodrigo, Refined blowup criteria and nonsymmetric blowup of an aggregation equation, Adv. Math., 220 (2009), 1717-1738.  doi: 10.1016/j.aim.2008.10.016.

[16]

D. Li, J. L. Rodrigo and X. Zhang, Exploding solutions for a nonlocal quadratic evolution problem, Rev. Mat. Iberoam., 26 (2010), 295-332.  doi: 10.4171/RMI/602.

[17]

T. Nagai, Behavior of solutions to a parabolic-elliptic system modelling chemotaxis, J. Korean Math. Soc., 37 (2000), 721-733. 

[18]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Ineq. Appl., 6 (2001), 37-55.  doi: 10.1155/S1025583401000042.

[19]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N. J. , 1970.

[20]

Y. Sugiyama, M. Yamamoto and K. Kato, Local and global solvability and blow up for the drift-diffusion equation with the fractional dissipation in the critical space, J. Diff. Eq., 258 (2015), 2983-3010.  doi: 10.1016/j.jde.2014.12.033.

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. 

[2]

P. Biler, I. Guerra and G. Karch, Large global-in-time solutions of the parabolic-parabolic Keller-Segel system on the plane, Comm. Pure Appl. Analysis, 14 (2015), 2117-2126.  doi: 10.3934/cpaa.2015.14.2117.

[3]

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.

[4]

P. Biler, G. Karch and P. Laurençot, Blowup of solutions to adiffusive aggregation model, Nonlinearity, 22 (2009), 1559-1568.  doi: 10.1088/0951-7715/22/7/003.

[5]

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.

[6]

P. Biler and W. A. Woyczyński, Global and exploding solutions of nonlocal quadratic evolution problems, SIAM J. Appl. Math., 59 (1999), 845-869.  doi: 10.1137/S0036139996313447.

[7]

P. Biler and J. Zienkiewicz, Existence of solutions for the Keller-Segel model of chemotaxis with measures as initial data, Bull. Polish Acad. Sci. Mathematics, 63 (2015), 41-51.  doi: 10.4064/ba63-1-6.

[8]

P. Biler, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations, 44 (2006), 32 pp. 

[9]

Y. Giga, T. Miyakawa and H. Osada, Two-dimensional Navier-Stokes flow with measures as initial vorticity, Arch. Rational Mech. Anal., 104 (1988), 223-250.  doi: 10.1007/BF00281355.

[10]

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.

[11]

H. Kozono and Y. Sugiyama, Local existence and finite time blow-up of solutions in the 2-D Keller-Segel system, J. Evol. Equ., 8 (2008), 353-378.  doi: 10.1007/s00028-008-0375-6.

[12]

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. 

[13]

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. Eq., 18 (2013), 1189-1208. 

[14]

D. Li and J. L. Rodrigo, Finite-time singularities of an aggregation equation in $\mathbb{R}^n$ with fractional dissipation, Comm. Math. Phys., 287 (2009), 687-703.  doi: 10.1007/s00220-008-0669-0.

[15]

D. Li and J. L. Rodrigo, Refined blowup criteria and nonsymmetric blowup of an aggregation equation, Adv. Math., 220 (2009), 1717-1738.  doi: 10.1016/j.aim.2008.10.016.

[16]

D. Li, J. L. Rodrigo and X. Zhang, Exploding solutions for a nonlocal quadratic evolution problem, Rev. Mat. Iberoam., 26 (2010), 295-332.  doi: 10.4171/RMI/602.

[17]

T. Nagai, Behavior of solutions to a parabolic-elliptic system modelling chemotaxis, J. Korean Math. Soc., 37 (2000), 721-733. 

[18]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Ineq. Appl., 6 (2001), 37-55.  doi: 10.1155/S1025583401000042.

[19]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N. J. , 1970.

[20]

Y. Sugiyama, M. Yamamoto and K. Kato, Local and global solvability and blow up for the drift-diffusion equation with the fractional dissipation in the critical space, J. Diff. Eq., 258 (2015), 2983-3010.  doi: 10.1016/j.jde.2014.12.033.

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