Radial blow-up in quasilinear Keller-Segel systems: approaching the full picture

Abstract The Neumann problem for the Keller-Segel system { u t = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( S ( u ) ∇ v ) , 0 = Δ v − μ + u , μ = − ∫ Ω u d x , is considered in n -dimensional balls Ω with n ⩾ 2 , with suitably regular and radially symmetric, radially nonincreasing initial data u 0 . The functions D and S are only assumed to belong to C 2 ( [ 0 , ∞ ) ) and to satisfy D > 0 and S ⩾ 0 on [ 0 , ∞ ) as well as S ( 0 ) = 0 ; in particular, diffusivities with arbitrarily fast decay are included. In this general context, it is shown that it is merely the asymptotic behavior as ξ → ∞ of the expression I ( ξ ) := S ( ξ ) ξ 2 n D ( ξ ) , ξ > 0 , which decides about the occurrence of blow-up: Namely, it is seen that • if lim ξ → ∞ I ( ξ ) = 0 , then any such solution is global and bounded, that • if lim sup ξ → ∞ I ( ξ ) < ∞ and ∫ Ω u 0 is suitably small, then the corresponding solution is global and bounded, and that • if lim inf ξ → ∞ I ( ξ ) > 0 , then at each appropriately large mass level m , there exist radial initial data u 0 such that ∫ Ω u 0 = m , and that the associated solution blows up either in finite or in infinite time. This especially reveals the presence of critical mass phenomena whenever lim ξ → ∞ I ( ξ ) ∈ ( 0 , ∞ ) exists.