A critical exponent in a quasilinear Keller-Segel system with arbitrarily fast decaying diffusivities accounting for volume-filling effects

The quasilinear Keller-Segel system { ut =del center dot ( D(u)del u)-del center dot ( S(u)del v), vt = del v - v + u, endowed with homogeneous Neumann boundary conditions is considered in a bounded domain Omega subset of R-n, n >= 3, with smooth boundary for sufficiently regular functions D and S satisfying D > 0 on [0,infinity), S > 0 on (0,8) and S(0) = 0. On the one hand, it is shown that if S/D satisfies the subcritical growth condition S(s)/D(s) <= Cs-alpha for all s >= 1 with some alpha < 2/n and C > 0, then for any sufficiently regular initial data there exists a global weak energy solution such that ess sup(t>) 0 parallel to u(t)parallel to (L p( Omega)) < infinity for some p > 2n/n+2. On the other hand, if S/D satisfies the supercritical growth condition S(s)/D(s) >= cs(alpha) for all s >= 1 with some alpha > 2/n and c > 0, then the nonexistence of a global weak energy solution having the boundedness property stated above is shown for some initial data in the radial setting. This establishes some criticality of the value alpha = 2/n for n >= 3, without any additional assumption on the behavior of D(s) as s -> infinity, in particular without requiring any algebraic lower bound for D. When applied to the Keller-Segel system with volume-filling effect for probability distribution functions of the type Q(s) = exp(-s(beta)), s >= 0, for global solvability the exponent beta = n-2/n is seen to be critical.