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{c}ot, Philippe}, 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}, number = {1}, year = {2010}, doi = {10.1016/j.anihpc.2009.11.016}, mrnumber = {2580517}, zbl = {1270.35377}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2009.11.016/} }
TY - JOUR AU - Cieślak, Tomasz AU - Laurençot, Philippe TI - Finite time blow-up for a one-dimensional quasilinear parabolic–parabolic chemotaxis system JO - Annales de l'I.H.P. Analyse non linéaire PY - 2010 SP - 437 EP - 446 VL - 27 IS - 1 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2009.11.016/ DO - 10.1016/j.anihpc.2009.11.016 LA - en ID - AIHPC_2010__27_1_437_0 ER -
%0 Journal Article %A Cieślak, Tomasz %A Laurençot, Philippe %T Finite time blow-up for a one-dimensional quasilinear parabolic–parabolic chemotaxis system %J Annales de l'I.H.P. Analyse non linéaire %D 2010 %P 437-446 %V 27 %N 1 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2009.11.016/ %R 10.1016/j.anihpc.2009.11.016 %G en %F 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) no. 1, pp. 437-446. doi : 10.1016/j.anihpc.2009.11.016. http://www.numdam.org/articles/10.1016/j.anihpc.2009.11.016/
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