Finite time blow-up for a one-dimensional quasilinear parabolic–parabolic chemotaxis system
Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 1, pp. 437-446.

Finite time blow-up is shown to occur for solutions to a one-dimensional quasilinear parabolic–parabolic chemotaxis system as soon as the mean value of the initial condition exceeds some threshold value. The proof combines a novel identity of virial type with the boundedness from below of the Liapunov functional associated to the system, the latter being peculiar to the one-dimensional setting.

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     title = {Finite time blow-up for a one-dimensional quasilinear parabolic{\textendash}parabolic chemotaxis system},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {437--446},
     publisher = {Elsevier},
     volume = {27},
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Cieślak, Tomasz; Laurençot, Philippe. Finite time blow-up for a one-dimensional quasilinear parabolic–parabolic chemotaxis system. Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 1, pp. 437-446. doi : 10.1016/j.anihpc.2009.11.016. http://archive.numdam.org/articles/10.1016/j.anihpc.2009.11.016/

[1] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, H. Schmeisser, H. Triebel (ed.), Function Spaces, Differential Operators, Nonlinear Analysis, Teubner-Texte Math. vol. 133, Teubner, Stuttgart (1993), 9-126 | MR

[2] P. Biler, T. Nadzieja, Existence and nonexistence of solutions for a model of gravitational interaction of particles, I, Colloq. Math. 66 (1994), 319-334 | EuDML | MR | Zbl

[3] A. Blanchet, J.A. Carrillo, Ph. Laurençot, Critical mass for a Patlak–Keller–Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations 35 (2009), 133-168 | MR | Zbl

[4] Th. Cazenave, Semilinear Schrödinger Equations, Courant Lect. Notes Math. vol. 10, Amer. Math. Soc., Providence (2003) | MR | Zbl

[5] T. Cieślak, Ph. Laurençot, Finite time blow-up for radially symmetric solutions to a critical quasilinear Smoluchowski–Poisson system, C. R. Acad. Sci. Paris Sér. I 347 (2009), 237-242 | MR | Zbl

[6] T. Cieślak, M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity 21 (2008), 1057-1076 | MR | Zbl

[7] H. Gajewski, K. Zacharias, Global behaviour of a reaction–diffusion system modelling chemotaxis, Math. Nachr. 195 (1998), 77-114 | MR | Zbl

[8] M.A. Herrero, J.J.L. Velázquez, Singularity patterns in a chemotaxis model, Math. Ann. 306 (1996), 583-623 | EuDML | MR | Zbl

[9] M.A. Herrero, J.J.L. Velázquez, Chemotactic collapse for the Keller–Segel model, J. Math. Biol. 35 (1996), 177-194 | MR | Zbl

[10] M.A. Herrero, J.J.L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Super. Pisa Cl. Sci. 24 (1997), 633-683 | EuDML | Numdam | MR | Zbl

[11] D. Horstmann, Lyapunov functions and L p -estimates for a class of reaction–diffusion systems, Colloq. Math. 87 (2001), 113-127 | EuDML | MR | Zbl

[12] D. Horstmann, On the existence of radially symmetric blow-up solutions for the Keller–Segel model, J. Math. Biol. 44 (2002), 463-478 | MR | Zbl

[13] D. Horstmann, From 1970 until present: the Keller–Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein. 105 (2003), 103-165 | MR | Zbl

[14] D. Horstmann, G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math. 12 (2001), 159-177 | MR | Zbl

[15] D. Horstmann, M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations 215 (2005), 52-107 | MR | Zbl

[16] W. Jäger, S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc. 329 (1992), 819-824 | MR | Zbl

[17] E.F. Keller, L.A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970), 399-415 | Zbl

[18] T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl. 5 (1995), 581-601 | MR | Zbl

[19] T. Nagai, Behavior of solutions to a parabolic–elliptic system modelling chemotaxis, J. Korean Math. Soc. 37 (2000), 721-733 | MR | Zbl

[20] T. Nagai, Blowup of nonradial solutions to parabolic–elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl. 6 (2001), 37-55 | EuDML | MR | Zbl

[21] T. Nagai, T. Senba, K. Yoshida, Application of the Trudinger–Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac. 40 (1997), 411-433 | MR | Zbl

[22] T. Senba, T. Suzuki, Parabolic system of chemotaxis: blowup in a finite and the infinite time, Methods Appl. Anal. 8 (2001), 349-368 | MR | Zbl

[23] T. Senba, T. Suzuki, A quasi-linear parabolic system of chemotaxis, Abstr. Appl. Anal. (2006), Article ID 23061, 1–21 | EuDML | MR

[24] Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller–Segel systems, Differential Integral Equations 19 (2006), 841-876 | MR | Zbl

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