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 $$ \left\{\begin{array}{ll} u_t= Δu - \nabla \cdot (u\nabla v), & x\in Ω, \, t>0, 0=Δv - μ(t)+w, & x\in Ω, \, t>0, τw_t + δw = u, & x\in Ω, \, t>0, \end{array} \right. \qquad \qquad (\star) $$ in the unit disk $Ω:=B_1(0)\subset \R^2$, where $δ\ge 0$ and $τ>0$ 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 $τ\to 0$, 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 $δ>0$ and $\io u_0<8πδ$, the solution remains uniformly bounded, whereas for any choice of $δ\ge 0$ and $m>8πδ$, 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.