Finite-time blow-up in low-dimensional Keller–Segel systems with logistic-type superlinear degradation

We consider radially symmetric solutions of the Keller–Segel system with generalized logistic source given by u t = Δ u - ∇ · ( u ∇ v ) + λ u - μ u κ , 0 = Δ v - v + u , ( ⋆ ) under homogeneous Neumann boundary conditions in the ball Ω = B R ( 0 ) ⊂ R n for n ≥ 3 and R > 0 , where λ ∈ R , μ > 0 and κ > 1 . Under the assumption that κ < 7 6 if n ∈ { 3 , 4 } , 1 + 1 2 ( n - 1 ) if n ≥ 5 , a condition on the initial data is derived which is seen to be sufficient to ensure the occurrence of finite-time blow-up for the corresponding solution of ( ⋆ ). Moreover, this criterion is shown to be mild enough so as to allow for the conclusion that in fact any positive continuous radial function on Ω ¯ is the limit in L 1 ( Ω ) of a sequence ( u 0 k ) k ∈ N of continuous radial initial data which are such that for each k ∈ N the associated initial-boundary value problem for ( ⋆ ) exhibits a finite-time explosion phenomenon in the above sense. In particular, this apparently provides the first rigorous detection of blow-up in a superlinearly dampened but otherwise essentially original Keller–Segel system in the physically relevant three-dimensional case.