The effects of cross-diffusion and logistic source on the boundedness of solutions to a pursuit-evasion model

We study the following quasilinear pursuit-evasion model:

$$\{\begin{array}{ll}{u}_t=\mathrm{\Delta }u-\chi \mathrm{\nabla }\cdot (u(u+1{)}^{\alpha }\mathrm{\nabla }w)+u({\lambda }_1-{\mu }_1{u}^{r_1-1}+av),& x\in \mathrm{\Omega },t>0,\\ v_t=\mathrm{\Delta }v+\xi \mathrm{\nabla }\cdot (v(v+1{)}^{\beta }\mathrm{\nabla }z)+v({\lambda }_2-{\mu }_2{v}^{r_2-1}-bu),& x\in \mathrm{\Omega },t>0,\\ 0=\mathrm{\Delta }w-w+v,& x\in \mathrm{\Omega },t>0,\\ 0=\mathrm{\Delta }z-z+u,& x\in \mathrm{\Omega },t>0,\end{array}$$

in a smooth and bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n(n\ge 1),$ where $a,b,\chi ,\xi ,{\lambda }_1,{\lambda }_2,{\mu }_1,{\mu }_2>0,$ $\alpha ,\beta \in \mathbb{R},$ and $r_1,r_2>1.$ When $r_1>max\{1,1+\alpha \},r_2>max\{1,1+\beta \},$ it has been proved that if $min\{(r_1-1)(r_2-\beta -1),(r_1-\alpha -1)(r_2-\beta -1)\}>\frac{(n-2{)}_{+}}{n},$ then for some suitable nonnegative initial data $u_0$ and $v_0,$ the system admits a unique globally classical solution which is bounded in $\mathrm{\Omega }\times (0,\mathrm{\infty })$.