×

Local criteria for blowup in two-dimensional chemotaxis models. (English) Zbl 1360.35290

Summary: 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.

MSC:

35Q92 PDEs in connection with biology, chemistry and other natural sciences
35B44 Blow-up in context of PDEs
35K55 Nonlinear parabolic equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] P. Biler, The Cauchy problem and self-similar solutions for a nonlinear parabolic equation,, Studia Math., 114, 181 (1995) · Zbl 0829.35044
[2] P. Biler, Large global-in-time solutions of the parabolic-parabolic Keller-Segel system on the plane,, Comm. Pure Appl. Analysis, 14, 2117 (2015) · Zbl 1326.35398 · doi:10.3934/cpaa.2015.14.2117
[3] P. Biler, Blowup of solutions to generalized Keller-Segel model,, J. Evol. Equ., 10, 247 (2010) · Zbl 1239.35177 · doi:10.1007/s00028-009-0048-0
[4] P. Biler, Blowup of solutions to a diffusive aggregation model,, Nonlinearity, 22, 1559 (2009) · Zbl 1177.35188 · doi:10.1088/0951-7715/22/7/003
[5] P. Biler, Optimal criteria for blowup of radial and \(N\)-symmetric solutions of chemotaxis systems,, Nonlinearity, 28, 4369 (2015) · Zbl 1329.35316 · doi:10.1088/0951-7715/28/12/4369
[6] P. Biler, Global and exploding solutions of nonlocal quadratic evolution problems,, SIAM J. Appl. Math., 59, 845 (1999) · Zbl 0940.35035 · doi:10.1137/S0036139996313447
[7] P. Biler, Existence of solutions for the Keller-Segel model of chemotaxis with measures as initial data,, Bull. Polish Acad. Sci. Mathematics, 63, 41 (2015) · Zbl 1326.35399 · doi:10.4064/ba63-1-6
[8] A. Blanchet, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions,, Electron. J. Differential Equations, 44 (2006) · Zbl 1112.35023
[9] Y. Giga, Two-dimensional Navier-Stokes flow with measures as initial vorticity,, Arch. Rational Mech. Anal., 104, 223 (1988) · Zbl 0666.76052 · doi:10.1007/BF00281355
[10] G. Karch, Blow-up versus global existence of solutions to aggregation equations,, Appl. Math. (Warsaw), 38, 243 (2011) · Zbl 1233.35046 · doi:10.4064/am38-3-1
[11] H. Kozono, Local existence and finite time blow-up of solutions in the 2-D Keller-Segel system,, J. Evol. Equ., 8, 353 (2008) · Zbl 1162.35040 · doi:10.1007/s00028-008-0375-6
[12] M. Kurokiba, Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type,, Differ. Integral Equ., 16, 427 (2003) · Zbl 1161.35432
[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, 1189 (2013) · Zbl 1284.35226
[14] D. Li, Finite-time singularities of an aggregation equation in \(\mathbbR^n\) with fractional dissipation,, Comm. Math. Phys., 287, 687 (2009) · Zbl 1178.35083 · doi:10.1007/s00220-008-0669-0
[15] D. Li, Refined blowup criteria and nonsymmetric blowup of an aggregation equation,, Adv. Math., 220, 1717 (2009) · Zbl 1168.35037 · doi:10.1016/j.aim.2008.10.016
[16] D. Li, Exploding solutions for a nonlocal quadratic evolution problem,, Rev. Mat. Iberoam., 26, 295 (2010) · Zbl 1195.35182 · doi:10.4171/RMI/602
[17] T. Nagai, Behavior of solutions to a parabolic-elliptic system modelling chemotaxis,, J. Korean Math. Soc., 37, 721 (2000) · Zbl 0962.35026
[18] T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains,, J. Ineq. Appl., 6, 37 (2001) · Zbl 0990.35024 · doi:10.1155/S1025583401000042
[19] E. M. Stein, <em>Singular Integrals and Differentiability Properties of Functions</em>,, Princeton Mathematical Series (1970) · Zbl 0207.13501
[20] Y. Sugiyama, Local and global solvability and blow up for the drift-diffusion equation with the fractional dissipation in the critical space,, J. Diff. Eq., 258, 2983 (2015) · Zbl 1321.35078 · doi:10.1016/j.jde.2014.12.033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.