Boundedness in a parabolic-elliptic chemotaxis system with nonlinear diffusion and sensitivity and logistic source

In this paper, we study the zero-flux chemotaxis-system (Formula presented.) where Ω is a bounded and smooth domain of ℝn, n≥1, and where m ∈ ℝ, k,μ>0 and α≤1. For any v≥0, the chemotactic sensitivity function is assumed to behave as the prototype χ(v)=χ0/(1+av)2, with a≥0 and χ0>0. We prove that for any nonnegative and sufficiently regular initial data u(x,0), the corresponding initial-boundary value problem admits a unique global bounded classical solution if α<1; indeed, for α=1, the same conclusion is obtained provided μ is large enough. Finally, we illustrate the range of dynamics present within the chemotaxis system in 1, 2, and 3 dimensions by means of numerical simulations.