Boundedness enforced by mildly saturated conversion in a chemotaxis-May-Nowak model for virus infection

Abstract

We study the system \begin{align*} \label{prob:star} \tag{$\star$} \begin{cases} u_t = Δu - \nabla \cdot (u \nabla v) - u - f(u) w + κ, \\ v_t = Δv - v + f(u) w, \\ w_t = Δw - w + v, \end{cases} \end{align*} which models the virus dynamics in an early stage of an HIV infection, in a smooth, bounded domain $Ω\subset \mathbb R^n, n \in \mathbb N,$ for a parameter $κ\ge 0$ and a given function $f \in C^1([0, \infty))$ satisfying $f \ge 0$, $f(0) = 0$ and $f(s) \le K_f s^α$ for all $s \ge 1$, some $K_f \gt 0$ and $α\in \mathbb R$. We prove that whenever \begin{align*} α\lt \frac2n, \end{align*} solutions to \eqref{prob:star} exist globally and are bounded. The proof mainly relies on smoothing estimates for the Neumann heat semigroup and (in the case $α\gt 1$) on a functional inequality. Furthermore, we provide some indication why the exponent $\frac2n$ could be essentially optimal.