Reduction of critical mass in a chemotaxis system by external application of a chemoattractant

Tello, José Ignacio ; Winkler, Michael

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 4, pp. 833-862

Résumé

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

$$\left\{\begin{array}{cc}{u}_t=\Delta u-\nabla ·\left(u\nabla v\right),\hfill & x\in {ℝ}^2,t>0,\hfill \\ 0=\Delta v+u+f_0·\delta \left(x\right),\hfill & x\in {ℝ}^2,t>0,\hfill \end{array}\right.(☆)$$

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 $8\pi -2f_0$ by an application of the signal substance in the above way. Indeed, it is proved that whenever $f_0>0$ and $u_0\neg \equiv 0$, a measure-valued global-in-time weak solution can be constructed which blows up at $x=0$ immediately. Now if $\mu <8\pi -2f_0$ then this solution satisfies $u(x,t)\le {C\left(\tau \right)\left|x\right|}^{-\frac{f_0}{2\pi }}$ for $t>\tau >0$ and $\left|x\right|<1$ 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, $u(·,t)$ converges to $\mu ·\delta$ as $t\to \infty$ in the sense of Radon measures.

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Publié le : 2019-02-21

Classification : 35K40, 35B44, 35K61