Global in Time and Bounded Solutions to a Parabolic–Elliptic Chemotaxis System with Nonlinear Diffusion and Signal-Dependent Sensitivity

Abstract

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 $$\varOmega $$ being a bounded and smooth domain of $$\mathbb {R}^n$$ , $$n\ge 1$$ , and where $$m,\alpha \in \mathbb {R}$$ , with $$\alpha \le \max \{m,\frac{m+1}{2}\}$$ . Additionally, $$0<\chi \in C^1((0,\infty ))$$ obeys the inequality $$\chi (s)\le \frac{\chi _0}{s^k}$$ , for some $$\chi _0>0, k\ge 1$$ 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 ( $$\Diamond $$ ) admits a unique globally bounded classical solution, provided some smallness assumptions on $$\chi _0$$ 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 ( $$\Diamond $$ ) may lead to complementary conclusions.