Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production

Abstract

We study the Neumann initial-boundary problem for the chemotaxis system $$\{\begin{array}{llcc}{u}_t=\Delta u-\nabla \cdot (u\nabla v),& x\in \Omega ,t>0,0=\Delta v-\mu (t)+w,& x\in \Omega ,t>0,\tau w_t+\delta w=u,& x\in \Omega ,t>0,\end{array}(\star )$$ in the unit disk $\Omega :=B_1(0)\subset {\mathbb{R}}^2$, where $\delta \ge 0$ and $\tau >0$ are given parameters and $μ(t):=\mint_Ωw(x,t)dx$, $t>0$. It is shown that this problem exhibits a novel type of critical mass phenomenon with regard to the formation of singularities, which drastically differs from the well-known threshold property of the classical Keller-Segel system, as obtained upon formally taking $\tau \to 0$, in that it refers to blow-up in infinite time rather than in finite time: Specifically, it is first proved that for any sufficiently regular nonnegative initial data $u_0$ and $w_0$, ($\star$) possesses a unique global classical solution. In particular, this shows that in sharp contrast to classical Keller-Segel-type systems reflecting immediate signal secretion by the cells themselves, the indirect mechanism of signal production in ($\star$) entirely rules out any occurrence of blow-up in finite time. However, within the framework of radially symmetric solutions it is next proved that whenever $\delta >0$ and $\io u_0<8πδ$, the solution remains uniformly bounded, whereas for any choice of $\delta \ge 0$ and $m>8\pi \delta$, one can find initial data such that $\io u_0=m$, and such that for the corresponding solution we have \bas \|u(\cdot,t)\|_{L^\infty(Ω)} \to \infty \qquad \mbox{as} t\to\infty.

37 pages, to appear in Journal of the European Mathematical Society