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 $\Omega \subset {\mathbb{R}}^n,n\in \mathbb{N},$ for a parameter $\kappa \ge 0$ and a given function $f\in C^1([0,\infty ))$ satisfying $f\ge 0$, $f(0)=0$ and $f(s)\le K_f{s}^{\alpha }$ for all $s\ge 1$, some $K_f>0$ and $\alpha \in \mathbb{R}$. 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 $\alpha >1$) on a functional inequality. Furthermore, we provide some indication why the exponent $\frac{2}{n}$ could be essentially optimal.

12 pages