References
- Winkler M. Blow-up profiles and life beyond blow-up in the fully parabolic Keller–Segel system. J Anal Math, to appear.
- Keller EF, Segel LA. Traveling bands of chemotactic bacteria: a theoretical analysis. J Theor Biol. 1971;30(2):235–248.
- Hillen T, Painter KJ. A user's guide to PDE models for chemotaxis. J Math Biol. 2009;58(1-2):183–217.
- Painter K, Hillen T. Volume-filling and quorum-sensing in models for chemosensitive movement. Can Appl Math Q;2002;10(4):501–544.
- Wrzosek D. Volume filling effect in modelling chemotaxis. Math Model Nat Phenom2010;5(1):123–147.
- Fu X, Tang L-H, Liu C, et al. Stripe formation in bacterial systems with density-suppressed motility. Phys Rev Lett. 2012;108(19):198102.
- Leyva JF, Málaga C, Plaza RG. The effects of nutrient chemotaxis on bacterial aggregation patterns with non-linear degenerate cross diffusion. Physica A. 2013;392(22):5644–5662.
- Kalinin YV, Jiang L, Tu Y, et al. Logarithmic sensing in Escherichia coli bacterial chemotaxis. Biophysical J. 2009;96(6):2439–2448.
- Bellomo N, Bellouquid A, Tao Y, et al. Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues. Math Models Methods Appl Sci. 2015;25(9):1663–1763.
- Lankeit J. Infinite time blow-up of many solutions to a general quasilinear parabolic–elliptic Keller–Segel system. Discrete Contin Dyn Syst S. 2020;13(2):233–255.
- Winkler M, Djie KC. Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect. Nonlinear Anal Theory Methods Appl. 2010;72(2):1044–1064.
- Horstmann D, Winkler M. Boundedness vs blow-up in a chemotaxis system. J Differ Equ. 2005;215(1):52–107.
- Ishida S, Seki K, Yokota T. Boundedness in quasilinear Keller–Segel systems of parabolic-parabolic type on non-convex bounded domains. J Differ Equ. 2014;256(8):2993–3010.
- Tao Y, Winkler M. Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with subcritical sensitivity. J Differ Equ. 2012;252(1):692–715.
- Calvez V, Carrillo JA. Volume effects in the Keller–Segel model: energy estimates preventing blow-up. J Math Pures Appl. 2006;86(2):155–175.
- Cieślak T, Winkler M. Finite-time blow-up in a quasilinear system of chemotaxis. Nonlinearity. 2008;21(5):1057–1076.
- Kowalczyk R, Szymańska Z. On the global existence of solutions to an aggregation model. J Math Anal Appl. 2008;343(1):379–398.
- Senba T, Suzuki T. A quasi-linear parabolic system of chemotaxis. Abstr Appl Anal. 2006;2006:1–21.
- Cieślak T, Winkler M. Global bounded solutions in a two-dimensional quasilinear Keller–Segel system with exponentially decaying diffusivity and subcritical sensitivity. Nonlinear Anal Real World Appl. 2017;35:1–19.
- Winkler M. Does a ‘volume-filling effect’ always prevent chemotactic collapse? Math Meth Appl Sci. 2009;33:12–24.
- Winkler M. Global existence and slow grow-up in a quasilinear Keller–Segel system with exponentially decaying diffusivity. Nonlinearity. 2017;30(2):735–764.
- Winkler M. Global classical solvability and generic infinite-time blow-up in quasilinear Keller–Segel systems with bounded sensitivities. J Differ Equ. 2019;266(12):8034–8066.
- Cieślak T, Stinner Ch. Finite-time blowup and global-in-time unbounded solutions to a parabolic–parabolic quasilinear Keller–Segel system in higher dimensions. J Differ Equ. 2012;252(10):5832–5851.
- Cieślak T, Stinner Ch. Finite-time blowup in a supercritical quasilinear parabolic–parabolic Keller–Segel system in dimension 2. Acta Appl Math. 2014;129(1):135–146.
- Cieślak T, Stinner Ch. New critical exponents in a fully parabolic quasilinear Keller–Segel system and applications to volume filling models. J Differ Equ. 2015;258(6):2080–2113.
- Cieślak T, Laurençot P. Finite time blow-up for a one-dimensional quasilinear parabolic–parabolic chemotaxis system. Ann Inst Henri Poincare (C) Nonlinear Anal. 2010;27(1):437–446.
- Senba T, Suzuki T. Chemotactic collapse in a parabolic–elliptic system of mathematical biology. Adv Differ Equ. 2001;6(1):21–50.
- Herrero MA, Velázquez JJL. A blow-up mechanism for a chemotaxis model. Ann Scuola Norm Sup Pisa Cl Sci (4). 1997;24(4):633–683.
- Nagai T, Senba T, Suzuki T. Chemotactic collapse in a parabolic system of mathematical biology. Hiroshima Math J. 2000;30(3):463–497.
- Souplet Ph, Winkler M. Blow-up profiles for the parabolic-elliptic Keller–Segel system in dimensions n≥3. Commun Math Phys. 2018;367:665–681.
- Fuest M. Blow-up profiles in quasilinear fully parabolic Keller–Segel systems. Nonlinearity. 2020;33(5):2306–2334.
- Freitag M. Blow-up profiles and refined extensibility criteria in quasilinear Keller–Segel systems. J Math Anal Appl. 2018;463(2):964–988.
- Triebel H. Interpolation theory, function spaces, differential operators. Amsterdam: North-Holland Pub. Co; 1978. (North-Holland mathematical library).
- Henry D. Geometric theory of semilinear parabolic equations. Berlin, Heidelberg: Springer Berlin Heidelberg; 1981. (Lecture Notes in Mathematics; vol. 840).
- Mora X. Semilinear parabolic problems define semiflows on Ck spaces. Trans Am Math Soc. 1983;278(1):21–55.
- Winkler M. Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system. J Math Pures Appl. 2013;100(5):748–767.
- Wang Y, Winkler M, Xiang Z. Global classical solutions in a two-dimensional chemotaxis – Navier–Stokes system with subcritical sensitivity. Ann SCUOLA Norm Super - Cl Sci. 2018;18(2):421–466.