Global boundedness and finite time blow-up of solutions for a quasilinear chemotaxis-May-Nowak model

In this paper, we introduce the nonlinear diffusion term $\mathrm{\nabla }\cdot (D(u)\mathrm{\nabla }u)$ into the chemotaxis-May-Nowak model to investigate the effects of $D(u)$ and chemotaxis on the global existence, boundedness, and finite time blow-up of solutions. Here, $D(u)$ generalizes the prototype $(1+u{)}^{m-1}$ with $m\in \mathbb{R}$. For the parabolic-elliptic-parabolic case, if $m>2+\frac{n}{2}-\frac{2}{n}$ when $n\ge 3$ and $m>\frac{3}{2}$ when $n=2$, then all solutions exist globally and remain bounded, whereas if $n\in \{2,3\}$ and $m<1$, finite time blow-up occurs when $\mathrm{\Omega }$ is a ball and the initial data are radially symmetric. For the fully parabolic case, if $m>1+\frac{n}{2}-\frac{2}{n}$, then all solutions exist globally and remain bounded.