Approaching optimality in blow-up results for Keller–Segel systems with logistic-type dampening

Nonnegative solutions of the Neumann initial-boundary value problem for the chemotaxis system [Figure not available: see fulltext.] in smooth bounded domains Ω ⊂ R ⁿ, n≥ 1 , are known to be global in time if λ≥ 0 , μ> 0 and κ> 2. In the present work, we show that the exponent κ= 2 is actually critical in the four- and higher dimensional setting. More precisely, if n≥4,κ∈(1,2)andμ>0orn≥5,κ=2andμ∈(0,n-4n),for balls Ω ⊂ R ⁿ and parameters λ≥ 0 , m> 0 , we construct a nonnegative initial datum u∈ C(Ω ¯) with ∫ _Ωu= m for which the corresponding solution (u, v) of (⋆) blows up in finite time. Moreover, in 3D, we obtain finite-time blow-up for κ∈(1,32) (and λ≥ 0 , μ> 0). As the corner stone of our analysis, for certain initial data, we prove that the mass accumulation function w(s,t)=∫0snρn-1u(ρ,t)dρ fulfills the estimate ws≤ws. Using this information, we then obtain finite-time blow-up of u by showing that for suitably chosen initial data, s and γ, the function ϕ(t)=∫0s0s-γ(s0-s)w(s,t) cannot exist globally.