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.
@article{AIHPC_2010__27_1_437_0, author = {Cie\'slak, Tomasz and Lauren\c cot, Philippe}, title = {Finite time blow-up for a one-dimensional quasilinear parabolic--parabolic chemotaxis system}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, volume = {27}, year = {2010}, pages = {437-446}, doi = {10.1016/j.anihpc.2009.11.016}, mrnumber = {2580517}, zbl = {1270.35377}, language = {en}, url = {http://dml.mathdoc.fr/item/AIHPC_2010__27_1_437_0} }
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, Tome 27 (2010) pp. 437-446. doi : 10.1016/j.anihpc.2009.11.016. http://gdmltest.u-ga.fr/item/AIHPC_2010__27_1_437_0/
[1] Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, , (ed.), Function Spaces, Differential Operators, Nonlinear Analysis, Teubner-Texte Math. vol. 133, Teubner, Stuttgart (1993), 9-126 | MR 1242579
,[2] Existence and nonexistence of solutions for a model of gravitational interaction of particles, I, Colloq. Math. 66 (1994), 319-334 | MR 1268074 | Zbl 0817.35041
, ,[3] Critical mass for a Patlak–Keller–Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations 35 (2009), 133-168 | MR 2481820 | Zbl 1172.35035
, , ,[4] Semilinear Schrödinger Equations, Courant Lect. Notes Math. vol. 10, Amer. Math. Soc., Providence (2003) | MR 2002047 | Zbl 1055.35003
,[5] 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 2537529 | Zbl 1175.35010
, ,[6] Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity 21 (2008), 1057-1076 | MR 2412327 | Zbl 1136.92006
, ,[7] Global behaviour of a reaction–diffusion system modelling chemotaxis, Math. Nachr. 195 (1998), 77-114 | MR 1654677 | Zbl 0918.35064
, ,[8] Singularity patterns in a chemotaxis model, Math. Ann. 306 (1996), 583-623 | MR 1415081 | Zbl 0864.35008
, ,[9] Chemotactic collapse for the Keller–Segel model, J. Math. Biol. 35 (1996), 177-194 | MR 1478048 | Zbl 0866.92009
, ,[10] A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Super. Pisa Cl. Sci. 24 (1997), 633-683 | Numdam | MR 1627338 | Zbl 0904.35037
, ,[11] Lyapunov functions and -estimates for a class of reaction–diffusion systems, Colloq. Math. 87 (2001), 113-127 | MR 1812147 | Zbl 0966.35022
,[12] On the existence of radially symmetric blow-up solutions for the Keller–Segel model, J. Math. Biol. 44 (2002), 463-478 | MR 1908133 | Zbl 1053.35064
,[13] From 1970 until present: the Keller–Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein. 105 (2003), 103-165 | MR 2013508 | Zbl 1071.35001
,[14] Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math. 12 (2001), 159-177 | MR 1931303 | Zbl 1017.92006
, ,[15] Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations 215 (2005), 52-107 | MR 2146345 | Zbl 1085.35065
, ,[16] On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc. 329 (1992), 819-824 | MR 1046835 | Zbl 0746.35002
, ,[17] Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970), 399-415 | Zbl 1170.92306
, ,[18] Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl. 5 (1995), 581-601 | MR 1361006 | Zbl 0843.92007
,[19] Behavior of solutions to a parabolic–elliptic system modelling chemotaxis, J. Korean Math. Soc. 37 (2000), 721-733 | MR 1783582 | Zbl 0962.35026
,[20] Blowup of nonradial solutions to parabolic–elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl. 6 (2001), 37-55 | MR 1887324 | Zbl 0990.35024
,[21] Application of the Trudinger–Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac. 40 (1997), 411-433 | MR 1610709 | Zbl 0901.35104
, , ,[22] Parabolic system of chemotaxis: blowup in a finite and the infinite time, Methods Appl. Anal. 8 (2001), 349-368 | MR 1904534 | Zbl 1056.92007
, ,[23] T. Senba, T. Suzuki, A quasi-linear parabolic system of chemotaxis, Abstr. Appl. Anal. (2006), Article ID 23061, 1–21 | MR 2211660
[24] 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 2263432 | Zbl 1212.35240
,