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 { u t = Δ u − χ ∇ ⋅ ( u ∇ v ) + λ u − μ u κ , x ∈ Ω , t > 0 , 0 = Δ v − m ( t ) + u , x ∈ Ω , t > 0 , in a ball Ω ⊂ R n , with parameters χ > 0 , λ ⩾ 0 , μ ⩾ 0 and κ > 1 , and m ( t ) : = 1 | Ω | ∫ Ω u ( x , t ) d x . It is shown that when n ⩾ 5 and κ < 3 2 + 1 2 n − 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.