Boundedness in a chemotaxis system with nonlinear signal production

This work deals with the zero-Neumann boundary problem to a fully parabolic chemotaxis system with a nonlinear signal production function f(s) fulfilling 0 <= f(s) <= Ks(alpha) for all s >= 0, where K and alpha are positive parameters. It is shown that whenever 0 < alpha < 2/n (where n denotes the spatial dimension) and under suitable assumptions on the initial data, this problem admits a unique global classical solution that is uniformly-in-time bounded in any spatial dimension. The proof is based on some a priori estimate techniques.