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

in the unit disk $\Omega :=B_1(0)\subset R^2$, where $\delta \ge 0$ and $\tau >0$ are given parameters and $\mu (t):=\frac{1}{|\Omega |}{\int }_{\Omega }w(x,t)dx$, $t>0$. 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 ${\int }_{\Omega }{u}_0<8\pi \delta$, 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 ${\int }_{\Omega }{u}_0=m$, and and the corresponding solution satisfies