Boundedness in a chemotaxis-May-Nowak model for virus dynamics with mildly saturated chemotactic sensitivity and conversion

This work studies a class of chemotaxis systems given by$ \begin{align*} \begin{cases} u_t = \Delta u-\nabla\cdot (uf(u)\nabla v)-u-g(u)w+\kappa, \\ v_t = \Delta v-v+g(u)w, \\ w_t = \Delta w-w+v, \end{cases} \end{align*} $under homogeneous Neumann boundary conditions in smooth bounded n-dimensional domains, where $ \kappa\geq 0 $, $ f\in C^2([0, \infty)) $ satisfies $ |f(s)|\leq K_f(1+s)^{-\alpha} $ for all $ s\geq 0 $ with some $ K_f>0 $ and $ \alpha\in\mathbb{R} $, and $ g\in C^1([0, \infty)) $ is a nonnegative function satisfying $ g(0) = 0 $ and $ g(s)\leq K_g(1+s^{\beta}) $ for all $ s\geq 1 $ with $ K_g>0 $ and $ \beta\in \mathbb{R} $. Under the assumption that $ \beta<\frac{2}{n} $ $ (n\geq 2) $, it is shown that there exists $ \alpha_0 = \alpha_0(\beta)>0 $ such that if $ \alpha>-\alpha_0 $, then the system has a global classical solution which is uniformly bounded. The case when $ n = 1 $ is also discussed. These improve the known results.