Boundedness in a Chemotaxis-May-Nowak model for virus dynamics with gradient-dependent flux limitation

Abstract

This paper investigates an extension of the May-Nowak ODE model for virus dynamics with gradient-dependent flux limitation of cross diffusion. In particular, we consider the associated no-flux initial–boundary value problem (0.1)$\left(\begin{array}{c}{u}_t=\Delta u-\nabla \cdot (uf\left({|\nabla v|}^2\right)\nabla v)+\kappa -u-uw,\\ v_t=\Delta v-v+uw,\\ w_t=\Delta w-w+v\end{array}\right)$in a smoothly bounded domain $\Omega \subset R^n(n\le 3)$, where the parameter $\kappa \ge 0$. The prototypical chemotactic sensitivity function $f\in C^2\left([0,\infty )\right)$ is given by $f\left(\xi \right)={(1+\xi )}^{-\alpha },\xi \ge 0$ with some $\alpha \in R$. It is proved that whenever $\left(\begin{array}{cc}\alpha \in R,& \mathrm{if}n=1,\\ \alpha >\frac{n-2}{2(n-1)},& \mathrm{if}n=\{2,3\},\end{array}\right)$global classical solutions to (0.1) exist and are uniformly bounded. Such result consists with that in [Winkler (2022), Proposition 1.2] when $n\le 3$, which shows that the effect of gradient-dependent flux limitation in weakening the cross-diffusion term remains unchanged even in the context of nonlinear signal production mechanism.