Critical mass for infinite time aggregation in a chemotaxis model with indirect signal production
We study the Neumann initial boundary problem for the chemotaxis system \left\\begin{array}{ll} 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} \right. \qquad \qquad \star) in the unit disk \Omega:=B_1(0)\subset \R^2$ where \delta\ge 0$ and \tau>0$ are given parameters and \mu(t)\mint_\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 \io u_0 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(\Omega) \to \infty \qquad \mbox{as} t\to\infty.