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 \{ {array}{ll} u_t= Δ u - ∇ · (u∇ v), & x∈ Ω, \, t>0, \\ 0=Δ v - μ(t)+w, & x∈ Ω, \, t>0, \\ τ w_t + δ w = u, & x∈ Ω, \, t>0, {array} . () in the unit disk Ω:=B_1(0)⊂ R^2 , where δ≥ 0 and τ>0 are given parameters and μ(t):=1/|Ω| ∫_Ω 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 τ→ 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 , ( ) 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 ( ) entirely rules out any occurrence of blow-up in finite time. However, within the framework of radially symmetric solutions it is next proved that \|u(·,t)\|L^∞(Ω) → ∞ as t→∞.