Abstract
The existence and nonexistence of global in time solutions is studied for a class of equations generalizing the chemotaxis model of Keller and Segel. These equations involve Lévy diffusion operators and general potential type nonlinear terms.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Biler P.: Local and global solvability of parabolic systems modelling chemotaxis. Adv. Math. Sci. Appl. 8, 715–743 (1998)
Biler P.: Existence and nonexistence of solutions for a model of gravitational interaction of particles. III. Colloq. Math. 68, 229–239 (1995)
Biler P.: The Cauchy problem and self-similar solutions for a nonlinear parabolic equation. Studia Math. 114, 181–205 (1995)
P. Biler, Radially symmetric solutions of a chemotaxis model in the plane – the supercritical case, 31–42, in: Parabolic and Navier-Stokes Equations, Banach Center Publications 81, Polish Acad. Sci., Warsaw, 2008.
Biler P., Brandolese L.: On the parabolic-elliptic limit of the doubly parabolic Keller–Segel system modelling chemotaxis. Studia Mathematica 193, 241–261 (2009)
Biler P., Cannone M., Guerra I., Karch G.: Global regular and singular solutions for a model of gravitating particles. Mathematische Annalen 330, 693–708 (2004)
Biler P., Funaki T., Woyczyński W.A.: Interacting particle approximation for nonlocal quadratic evolution problems. Probab. Math. Stat. 19, 267–286 (1999)
Biler P., Karch G., Woyczyński W.A.: Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws. Ann. Inst. H. Poincaré – Analyse non Linéaire 18, 613–637 (2001)
Biler P., Karch G., Woyczyński W.A.: Asymptotics for conservation laws involving Lévy diffusion generators. Studia Math. 148, 171–192 (2001)
Biler P., Woyczyński W.A.: Global and exploding solutions for nonlocal quadratic evolution problems. SIAM J. Appl. Math. 59, 845–869 (1998)
P. Biler, G. Wu, Two-dimensional chemotaxis models with fractional diffusion, Math. Methods Appl. Sciences 32 (2009), 112–126; DOI: 10.1002/mma.1036, (2008)
Blanchet A., Dolbeault J., Perthame B.: Two dimensional Keller–Segel model: Optimal critical mass and qualitative properties of the solutions. Electron. J. Diff. Eqns. 44, 1–33 (2006)
Brandolese L., Karch G.: Far field asymptotics of solutions to convection equation with anomalous diffusion. J. Evolution Equations 8, 307–326 (2008)
V. Calvez, B. Perthame, M. Sharifi tabar, Modified Keller–Segel system and critical mass for the log interaction kernel, Contemporary Math. 429, 45–62 (2007), Stochastic analysis and pde; Chen, Gui-Qiang (ed.) et al., AMS.
Corrias L., Perthame B., Zaag H.: Global solutions of some chemotaxis and angiogenesis systems in high space dimensions. Milan J. Math. 72, 1–29 (2004)
Droniou J., Imbert C.: Fractal first order partial differential equations. Arch. Rat. Mech. Anal. 182, 299–331 (2006)
Escudero C.: The fractional Keller–Segel model. Nonlinearity 19, 2909–2918 (2006)
Jacob N.: Pseudo-differential Operators and Markov Processes, vol. 1: Fourier analysis and semigroups. Imperial College Press, London (2001)
Jäger W., Luckhaus S.: On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Amer. Math. Soc. 329, 819–824 (1992)
Karch G.: Scaling in nolinear parabolic equations. J. Math. Anal. Appl. 234, 534–558 (1999)
Kozono H., Sugiyama Y.: Local existence and finite time blow-up of solutions in the 2-D Keller-Segel system. J. Evolution Equations 8, 353–378 (2008)
Kurokiba M., Ogawa T.: Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type. Differential Integral Equations 16, 427–453 (2003)
Lemarié-Rieusset P.-G.: Recent Development in the Navier–Stokes Problem. Chapman & Hall/CRC Press, Boca Raton (2002)
D. Li, J. Rodrigo, X. Zhang, Exploding solutions for a nonlocal quadratic evolution problem, 1–40, preprint on the webpage http://www.warwick.ac.uk/staff/J.Rodrigo/research.html
Y. Meyer, Wavelets, paraproducts and Navier–Stokes equations, Current developments in mathematics, 1996, Internat. Press, Cambridge, MA 02238-2872 (1999).
Nagai T.: Behavior of solutions to a parabolic-elliptic system modelling chemotaxis. J. Korean Math. Soc. 37, 721–732 (2000)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Biler, P., Karch, G. Blowup of solutions to generalized Keller–Segel model. J. Evol. Equ. 10, 247–262 (2010). https://doi.org/10.1007/s00028-009-0048-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00028-009-0048-0