A sufficient condition of sensitivity functions for boundedness of solutions to a parabolic-parabolic chemotaxis system

This paper deals with time-global solutions to the parabolic system {tau u(t) = Delta u - del. (u del chi(v)) in Omega x (0, infinity), v(t) = Delta v - v + u in Omega x (0, infinity) under the homogeneous Neumann boundary conditions in a bounded and convex domain Omega subset of R-n (n >= 2) with smooth boundary partial derivative Omega. Here tau is a positive parameter, chi is a smooth function on (0, infinity) satisfying chi' > 0 and (u(0), v(0)) is a pair of nonnegative initial data. We will consider the above system as a perturbation of a nonlocal parabolic equation and establish a sufficient condition of the sensitivity function. for the global existence of solutions under the assumption of smallness of the constant tau.