Global boundedness in a Chemotaxis-May-Nowak model for virus dynamics with logistic damping

Abstract

This paper investigates the following May--Nowak type model for viral infection in a bounded domain $\Omega \subset \mathbb{R}^n$, $n \ge 2$: $\begin{cases} u_t = \Delta u - \chi \nabla \cdot (u \nabla v) + \kappa - u - uw - \mu \dfrac{u^{1+\alpha}}{\ln^k(u+e)}, \\[1mm] \gamma v_t = \Delta v - v + uw, \\[1mm] w_t = \Delta w - w + v, \end{cases}$ where $\chi \in \mathbb{R}$, $\mu>0$, $k \in [0,1)$, $\alpha>0$, and $\gamma \in \{0,1\}$. We establish the global existence and uniform-in-time boundedness of classical solutions for suitably regular initial data under one of the following conditions: (A) $\gamma = 1$, $n = 2$, $k \in [0,1)$, $\alpha = 1$, and $\mu>0$; (B) $\gamma = 1$, $3 \le n \le 5$, $k = 0$, $\alpha = 1$, and $\mu $ is sufficiently large; (C) $\gamma = 1$, $n \ge 6$, $k = 0$, $\alpha>\frac{n-2}{4}$, and $\mu>0$; (D) $\gamma = 0$, $n \ge 2$, $k = 0$, $\alpha>\frac{n+2}{2}$, and $\mu>0$. In particular, in the physically relevant dimensions $n = 2,3$, both subquadratic and quadratic damping are sufficient to prevent blow-up. Moreover, in the fully parabolic case ($\gamma = 1$), our results improve upon recent findings by relaxing the condition $\alpha>\frac{n}{2}$ to weaker assumptions within the above parameter regimes.