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

$\left\{\begin{array}{cc}{u}_{t}=\Delta u-\nabla ·\left(u\nabla v\right),\hfill & x\in {ℝ}^{2},\phantom{\rule{4pt}{0ex}}t>0,\hfill \\ 0=\Delta v+u+{f}_{0}·\delta \left(x\right),\hfill & x\in {ℝ}^{2},\phantom{\rule{4pt}{0ex}}t>0,\hfill \end{array}\right\\left(☆\right)$

where ${f}_{0}>0$ and $\delta$ 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 $8\pi$ plays the role of a critical mass in ($☆$), in the sense that if the total mass $\mu :={\int }_{{ℝ}^{2}}{u}_{0}$ of the cells exceeds $8\pi$ then the solutions may blow up within finite time and collapse into a Dirac-type singularity, and that this does not occur when $\mu <8\pi$.

The present paper shows that this critical number is reduced to $8\pi -2{f}_{0}$ by an application of the signal substance in the above way. Indeed, it is proved that whenever ${f}_{0}>0$ and ${u}_{0}¬\equiv 0$, a measure-valued global-in-time weak solution can be constructed which blows up at $x=0$ immediately. Now if $\mu <8\pi -2{f}_{0}$ then this solution satisfies $u\left(x,t\right)\le {C\left(\tau \right)|x|}^{-\frac{{f}_{0}}{2\pi }}$ for $t>\tau >0$ and $|x|<1$ and hence does not blow up in ${L}_{\mathrm{loc}}^{p}\left({ℝ}^{2}\right)$ for any $1\le p<4\pi /{f}_{0}$. On the other hand, if $\mu >8\pi -{f}_{0}$ then the mass will asymptotically completely concentrate at the origin, that is, $u\left(·,t\right)$ converges to $\mu ·\delta$ as $t\to \infty$ 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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