Abstract
Qualitative properties of solutions blowing up in finite time are obtained for a degenerate parabolic–parabolic Keller–Segel system, the nonlinear diffusion being of porous medium type with an exponent smaller or equal to the critical one \(m_c:=2(N-1)/N\). In both cases, it is shown that only type II blow-up is possible, that is, blow-up at the same rate as backward self-similar solutions never occurs. Further information on the generation of singularities induced by mass concentration are given in the critical case.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Biler, P., Corrias, L., Dolbeault, J.: Large mass self-similar solutions of the parabolic–parabolic Keller–Segel model of chemotaxis. J. Math. Biol. 63, 1–32 (2011)
Blanchet, A., Laurençot, Ph: Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion. Commun. Pure Appl. Anal. 11, 47–60 (2012)
Cieślak, T., Stinner, C.: Finite-time blowup and global-in-time unbounded solutions to a parabolic–parabolic quasilinear Keller–Segel system in higher dimensions. J. Differ. Equ. 252, 5832–5851 (2012)
Cieślak, T., Stinner, C.: Finite-time blowup in a supercritical quasilinear parabolic–parabolic Keller–Segel system in dimension 2. Acta Appl. Math. 129, 135–146 (2014)
Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. In: Classics in Mathematics. Springer, Berlin (2001)
Herrero, M.A., Velázquez, J.J.L.: Singularity patterns in a chemotaxis model. Math. Ann. 306, 583–623 (1996)
Herrero, M.A., Velázquez, J.J.L.: A blow-up mechanism for a chemotaxis model. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 24, 633–683 (1997)
Horstmann, D., Winkler, M.: Boundedness vs. blow-up in a chemotaxis system. J. Differ. Equ. 215, 52–107 (2005)
Ishida, S., Yokota, T.: Blow-up in finite or infinite time for quasilinear degenerate Keller–Segel systems of parabolic–parabolic type. Discret. Contin. Dyn. Syst. Ser. B 18, 2569–2596 (2013)
Ishida, S., Ono, T., Yokota, T.: Possibility of the existence of blow-up solutions to quasilinear degenerate Keller–Segel systems of parabolic–parabolic type. Math. Methods Appl. Sci. 36, 745–760 (2013)
Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)
Lamberton, D.: Equations d’évolution linéaires associées à des semi-groupes de contractions dans les espaces \(L^p\). J. Funct. Anal. 72, 252–262 (1987)
Laurençot, P., Mizoguchi, N.: Finite time blowup for the parabolic–parabolic Keller–Segel system with critical diffusion. Ann. Inst. H. Poincaré Anal. Non Linéaire (pulished online 2015)
Lieberman, G.M.: Second Order Parabolic Differential Equations. World Scientific, Singapore (1996)
Mizoguchi, N.: A new proof to finite-time blowup and no infinite-time blowup in doubly parabolic Keller–Segel system (preprint)
Mizoguchi, N.: Type II blowup in the doubly parabolic Keller–Segel system in the two dimension (preprint)
Mizoguchi, N.: Nonexistence of type I blowup solutions to parabolic–parabolic Keller–Segel system (preprint)
Mizoguchi, N., Senba, T.: Refined asymptotics of blowup solutions to a simplified chemotaxis system (preprint)
Mizoguchi, N., Souplet, Ph: Nondegeneracy of blow-up points for the parabolic Keller–Segel system. Ann. Inst. H. Poincaré Anal. Non Linéaire 31, 851–875 (2014)
Mizoguchi, N., Winkler, M.: Blow-up in the two-dimensional parabolic Keller–Segel system (preprint)
Mizoguchi, N., Winkler, M.: Boundedness of global solutions in the two-dimensional parabolic Keller–Segel system (preprint)
Nagai, T., Senba, T., Suzuki, T.: Chemotactic collapse in a parabolic system of mathematical biology. Hiroshima Math. J. 30, 463–497 (2000)
Raphaël, P., Schweyer, R.: On the stability of critical chemotactic aggregation. Math. Ann. 359, 267–377 (2014)
Schweyer, R.: Stable blow-up dynamic for the parabolic–parabolic Patlak–Keller-Segel model (preprint). arXiv:1403.4975
Senba, T.: Type II blowup of solutions to a simplified Keller–Segel system in two dimensional domains. Nonlinear Anal. 66, 1817–1839 (2007)
Senba, T., Suzuki, T.: Applied Analysis: Mathematical Methods in Natural Science, 2nd edn. Imperial College Press, London (2011)
Sugiyama, Y.: On \(\varepsilon \)-regularity theorem and asymptotic behaviors of solutions for Keller–Segel systems. SIAM J. Math. Anal. 41, 1664–1692 (2009)
Sugiyama, Y.: \(\varepsilon \)-regularity theorem and its application to the blow-up solutions of Keller–Segel systems in higher dimensions. J. Math. Anal. Appl. 364, 51–70 (2010)
Sugiyama, Y., Kunii, H.: Global existence and decay properties for a degenerate Keller–Segel model with a power factor in drift term. J. Differ. Equ. 227, 333–364 (2006)
Suzuki, T., Takahashi, R.: Degenerate parabolic equation with critical exponent derived from the kinetic theory, III, \(\varepsilon \)-regularity. Differ. Integral Equ. 25, 223–250 (2012)
Tanabe, H.: Functional Analytic Methods for Partial Differential Equations. Dekker, New York (1997)
Winkler, M.: Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system. J. Math. Pures Appl. 100, 748–767 (2013)
Acknowledgments
We warmly thank the referee for carefully reading the manuscript and relevant remarks. Ishige was partially supported by the JSPS Grant-in-Aid for Scientific Research (A) (No. 15H02058). Mizoguchi was partially supported by the JSPS Grant-in-Aid for Scientific Research (B) (No. 26287021) and by the JSPS Grant-in-Aid for Scientific Research (S) (No. 23224003).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ishige, K., Laurençot, P. & Mizoguchi, N. Blow-up behavior of solutions to a degenerate parabolic–parabolic Keller–Segel system. Math. Ann. 367, 461–499 (2017). https://doi.org/10.1007/s00208-016-1400-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-016-1400-7