Global in time and bounded solutions to a parabolic–elliptic chemotaxis system with nonlinear diffusion and signal-dependent sensitivity

This paper deals with a system of two coupled partial differential equations arising in chemotaxis, involving nonlinear diffusion and nonlinear and signal-dependent sensitivity. Depending on the interplay between such nonlinearities, we establish the existence of global classical solutions which are uniformly bounded in time. Precisely, we study the zero-flux chemotaxis-system [Figure not available: see fulltext.]Ω being a bounded and smooth domain of R n , n≥ 1 , and where m, α∈ R, with α≤maxm,m+12. Additionally, 0 < χ∈ C 1 ((0 , ∞)) obeys the inequality χ(s)≤χ0sk, for some χ> 0 , k≥ 1 and all s> 0. We prove that for any nonnegative and properly regular initial data u(x, 0), the initial-boundary value problem associated to (◊) admits a unique globally bounded classical solution, provided some smallness assumptions on χ are satisfied. In addition, in this article we compare our results with those achieved in the recent paper (Wang et al. in J Differ Equ 263(5):2851–2873, 2017); we will emphasize how the employment of independent techniques used to solve problem (◊) may lead to complementary conclusions.