In this paper we study non-negative radially symmetric solutions of the parabolic-elliptic Keller-Segel system
where and is the Dirac distribution. This system describes the chemotactic movement of cells under the additional circumstance that an external application of a chemoattractant at a distinguished point is introduced.
It is known that without such an external source the number plays the role of a critical mass in (), in the sense that if the total mass of the cells exceeds then the solutions may blow up within finite time and collapse into a Dirac-type singularity, and that this does not occur when .
The present paper shows that this critical number is reduced to by an application of the signal substance in the above way. Indeed, it is proved that whenever and , a measure-valued global-in-time weak solution can be constructed which blows up at immediately. Now if then this solution satisfies for and and hence does not blow up in for any . On the other hand, if then the mass will asymptotically completely concentrate at the origin, that is, converges to as in the sense of Radon measures.
@article{ASNSP_2013_5_12_4_833_0, author = {Tello, Jos\'e Ignacio and Winkler, Michael}, title = {Reduction of critical mass in a chemotaxis system by external application of a chemoattractant}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {833--862}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 12}, number = {4}, year = {2013}, mrnumber = {3184571}, zbl = {1295.35136}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2013_5_12_4_833_0/} }
TY - JOUR AU - Tello, José Ignacio AU - Winkler, Michael TI - Reduction of critical mass in a chemotaxis system by external application of a chemoattractant JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2013 SP - 833 EP - 862 VL - 12 IS - 4 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2013_5_12_4_833_0/ LA - en ID - ASNSP_2013_5_12_4_833_0 ER -
%0 Journal Article %A Tello, José Ignacio %A Winkler, Michael %T Reduction of critical mass in a chemotaxis system by external application of a chemoattractant %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2013 %P 833-862 %V 12 %N 4 %I Scuola Normale Superiore, Pisa %U http://www.numdam.org/item/ASNSP_2013_5_12_4_833_0/ %G en %F ASNSP_2013_5_12_4_833_0
Tello, José Ignacio; Winkler, Michael. Reduction of critical mass in a chemotaxis system by external application of a chemoattractant. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 4, pp. 833-862. http://www.numdam.org/item/ASNSP_2013_5_12_4_833_0/
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