Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction

We study radially symmetric solutions of a class of chemotaxis systems generalizing the prototype{ut=Δu−χ∇⋅(u∇v)+λu−μuκ,x∈Ω,t>0,0=Δv−m(t)+u,x∈Ω,t>0, in a ball Ω⊂RnΩ⊂Rn, with parameters χ>0χ>0, λ⩾0λ⩾0, μ⩾0μ⩾0 and κ>1κ>1, and m(t):=1|Ω|∫Ωu(x,t)dx. It is shown that when n⩾5n⩾5 andκ<32+12n−2, then there exist initial data such that the smooth local-in-time solution of (⋆) blows up in finite time. This indicates that even superlinear growth restrictions may be insufficient to rule out a chemotactic collapse, as is known to occur in the corresponding system without any proliferation.