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 $\Omega \subset {\mathbb{R}}^n$, $n\ge 2$: $\{\begin{array}{l}{u}_t=\Delta u-\chi \nabla \cdot (u\nabla v)+\kappa -u-uw-\mu \frac{u^{1+\alpha }}{{\ln }^k(u+e)},\\ \gamma v_t=\Delta v-v+uw,\\ w_t=\Delta w-w+v,\end{array}$ 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.