Global classical solutions and stabilization for a class of competition models with density-dependent motility

In this paper, we consider the competition system$ \begin{align} \begin{cases} u_t = \nabla\cdot(\gamma(v)\nabla u-\chi(v)u\nabla v)+\mu_1 u(1-u-a_1v), &x\in\Omega, \, t>0, \\ v_t = \Delta v+\mu_2v(1-a_2u-v), &x\in\Omega, \, t>0, \end{cases} \end{align} $under homogeneous Neumann boundary conditions. $ \Omega\subset \mathbb{R}^n(n\geqslant1) $ is a smooth bounded domain, and $ \gamma\in C^3([0, +\infty)) $, $ \chi\in C^3([0, +\infty)) $, and $ \gamma>0 $ on $ [0, +\infty) $. It is shown that if the initial data is sufficiently smooth and nonnegative, then the problem possesses a unique global-in-time classical bounded solution. Furthermore, by the method of Lyapunov functionals, the global stability of steady states with convergence rates is established.