Global bounded solutions in a two-dimensional quasilinear Keller-Segel system with exponentially decaying diffusivity and subcritical sensitivity

The quasilinear chemotaxis system {ut = del center dot (D(u)del u) -del center dot(S(u)del v), v(t) = del v - v + u, is considered under homogeneous Neumann boundary conditions in a bounded domain Omega subset of R-2 with smooth boundary. It is shown that if DD and SS are sufficiently smooth nonnegative functions on [0,infinity) satisfying K(1)e(-beta-s) <= D(s) <= K(2)e(-beta+s) for all s >= 0 with some K-1 > 0, K-2 > 0, beta(+) > 0 and beta(-) >= beta(+) then whenever S satisfies the condition of subcritical growth relative to DD given by S(s)/D(s) <= K(3)s(alpha) for all s >= 0 with some K-3 > 0 and alpha epsilon (0,1), for all suitably regular nonnegative initial data the corresponding initial-boundary value problem for (star) possesses a global classical solution for which the component u is bounded in Omega x (0, infinity). (C) 2016 Elsevier Ltd. All rights reserved.