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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 & Continuous Dynamical Systems - A, 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[19]

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

[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.  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, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[19]

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

[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.  Google Scholar

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