Reduction of critical mass in a chemotaxis system by external application of a chemoattractant
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 12 (2013) no. 4, p. 833-862

In this paper we study non-negative radially symmetric solutions of the parabolic-elliptic Keller-Segel system

ut=Δu-·(uv),x2,t>0,0=Δv+u+f0·δ(x),x2,t>0,()

where f 0 >0 and δ 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 8π plays the role of a critical mass in (), in the sense that if the total mass μ:= 2 u 0 of the cells exceeds 8π then the solutions may blow up within finite time and collapse into a Dirac-type singularity, and that this does not occur when μ<8π.

The present paper shows that this critical number is reduced to 8π-2f 0 by an application of the signal substance in the above way. Indeed, it is proved that whenever f 0 >0 and u 0 ¬0, a measure-valued global-in-time weak solution can be constructed which blows up at x=0 immediately. Now if μ<8π-2f 0 then this solution satisfies u(x,t)C(τ)|x| -f 0 2π for t>τ>0 and |x|<1 and hence does not blow up in L loc p ( 2 ) for any 1p<4π/f 0 . On the other hand, if μ>8π-f 0 then the mass will asymptotically completely concentrate at the origin, that is, u(·,t) converges to μ·δ as t in the sense of Radon measures.

Published online : 2019-02-21
Classification:  35K40,  35B44,  35K61
@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},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 12},
     number = {4},
     year = {2013},
     pages = {833-862},
     zbl = {1295.35136},
     mrnumber = {3184571},
     language = {en},
     url = {http://www.numdam.org/item/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, Serie 5, Volume 12 (2013) no. 4, pp. 833-862. http://www.numdam.org/item/ASNSP_2013_5_12_4_833_0/

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