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) ≤ Ksαfor all s ≥ 0, where K and α are positive parameters. It is shown that whenever 0 < α <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.