2016/08/26 by Tao, Youshan, Winkler, Michael · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1608.07622
We study the Neumann initial-boundary problem for the chemotaxis system \ut= Δu - ∇ ⋅ (u∇ v), · amp; x∈ Ω, t · gt;0, 0=Δv - μ(t)+w, · amp; x∈ Ω, t · gt;0, τwt + δw = u, · amp; x∈ Ω, t · gt;0, . (⋆) in the unit disk Ω:=B1(0)⊂ \R2, where δ≥ 0 and τ>0 are given parameters and μ(t):=\mintΩw(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 u0 and w0, (⋆) 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 whenever δ>0 and \io u0<8πδ, the solution remains uniformly bounded, whereas for any choice of δ≥ 0 and m>8πδ, one can find initial data such that \io u0=m, and such that for the corresponding solution we have \bas ‖u(⋅,t)‖L^∞(Ω) → ∞ as t→∞.