Existence of global solutions and asymptotic stability behavior for the two-diffusion model with nonlinear diffusion

This paper focuses on the study of a quasilinear parabolic forager-exploiter system with logistic sources:$ \begin{cases} u_t = \nabla\cdot((u + 1)^m\nabla u)-\nabla\cdot(u\nabla w)+\lambda( u - u^k), & x \in \Omega, \ t > 0, \\v_t = \nabla\cdot((v + 1)^l\nabla v)-\nabla\cdot(v\nabla u)+\eta( v - v^h), & x \in \Omega, \ t > 0, \\ w_t = \Delta w-(u + v)w-\mu w + r(x, t), & x \in \Omega, \ t > 0, \end{cases} $where $\Omega$ is a bounded domain in $\mathbb{R}^n(n\geq1)$ with smooth boundary $\partial\Omega$. We assume initial data $(u_0, v_0, w_0)$ satisfy certain conditions and the non-negative function $r$ fulfills $r\in C^2(\Omega\times[0, +\infty))\cap L^{\infty}(\Omega\times(0, +\infty))$. It is shown that if $ m, l> 1 $ and $ k, h \geq 2 $, this system possesses a unique global bounded classical solution. Moreover, when additional hypothesis is imposed on $r(x, t)$, the asymptotic behavior can be investigated.