Boundedness in a fully parabolic chemotaxis system with strongly singular sensitivity

This paper presents global existence and boundedness of classical solutions to the fully parabolic chemotaxis system u(t) = Delta u - del.(u chi(v)del v), v(t) = Delta v-v+u with the strongly singular sensitivity function chi(v) such that 0 < chi(v) <= chi(0)/v(k) (chi(0) > 0, k > 1). As to the regular case 0 < chi(v) <= chi(0)/(1+alpha v)(k) (alpha > 0, chi(0) > 0, k > 1), it has been shown, by Winkler (2010), that the system has a unique global classical solution which is bounded in time, whereas this method cannot be directly applied to the singular case. In the present work, a uniform-in-time lower bound for v is established and builds a bridge between the regular case as in Winkler (2010) and the singular one. (C) 2014 Elsevier Ltd. All rights reserved.