Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity

We consider the chemotaxis system {u(t) = Delta u - del.(u chi(v)del v), x is an element of Omega, t > 0, u(t) = Delta v - v + u, x is an element of Omega, t > 0, under homogeneous Neumann boundary conditions in a smooth bounded domain Omega subset of R(n). The chemotactic sensitivity function is assumed to generalize the prototype chi(v) = chi o/(1+alpha v)(2), v >= 0. It is proved that no chemotactic collapse occurs in the sense that for any choice of nonnegative initial data u(.,0) is an element of C(0) ((Omega) over bar) and v(.,0) is an element of W(1, r) (Omega) (with some r > n), the corresponding initial-boundary value problem possesses a unique global solution that is uniformly bounded. (C) 2010 WILEY- VCH Verlag GmbH & Co. KGaA, Weinheim