Slow Grow-up in a Quasilinear Keller-Segel System

In a ball Omega = B-R(0) subset of R-n, n >= 2, the chemotaxis system {u(t) = del . (D(u)del u) - del. (uS(u)del v), 0 = Delta v - mu + u, mu = 1/Omega integral(Omega) u, (*) is considered under no-flux boundary conditions, with a focus on nonlinearities S is an element of C-2 ([0, infinity)) which exhibit super-algebraically fast decay in the sense that with some K-S > 0, beta is an element of [0, 1) and xi(0) > 0, S(xi) > 0 and S'(xi) <= -K-S xi(-beta) S(xi) for all xi >= xi(0). It is, inter alia, shown that if furthermore D is an element of C-2 ((0, infinity)) is positive and suitably small in relation to S by satisfying xi S(xi)/D(xi) >= K-SD xi(lambda) for xi >= xi(0) with some K-SD > 0 and lambda > 2/n, then throughout a considerably large set of initial data, (*) admits global classical solutions (u, v) fulfilling z(t)/C <= parallel to u(., t) parallel to (L infinity(Omega)) <= Cz(t) for all t > 0, with some C = C-(u,C- v) >= 1, where z denotes the solution of {z'(t) = z(2)(t) . S(z(t)), t > 0, z(0) = xi(0), which is seen to exist globally, and to satisfy z(t) -> +infinity as t -> infinity. As particular examples, exponentially and doubly exponentially decaying S are found to imply corresponding infinite-time blow-up properties in (*) at logarithmic and doubly logarithmic rates, respectively.