The Keller-Segel system with logistic growth and signal-dependent motility

The paper is concerned with the following chemotaxis system with nonlinear motility functions

$\begin{equation}\label{0-1}\begin{cases}u_t = \nabla \cdot (\gamma(v)\nabla u- u\chi(v)\nabla v)+\mu u(1-u), &x\in \Omega, ~~t>0, \\ 0 = \Delta v+ u-v, & x\in \Omega, ~~t>0, \\u(x, 0) = u_0(x), & x\in \Omega, \end{cases}~~~~(\ast)\end{equation}$

subject to homogeneous Neumann boundary conditions in a bounded domain $ \Omega\subset \mathbb{R}^2 $ with smooth boundary, where the motility functions $ \gamma(v) $ and $ \chi(v) $ satisfy the following conditions

● $ (\gamma, \chi)\in [C^2[0, \infty)]^2 $ with $ \gamma(v)>0 $ and $ \frac{|\chi(v)|^2}{\gamma(v)} $ is bounded for all $ v\geq 0 $.

By employing the method of energy estimates, we establish the existence of globally bounded solutions of ($\ast$) with $ \mu>0 $ for any $ u_0 \in W^{1, \infty}(\Omega) $ with $ u_0 \geq (\not\equiv) 0 $. Then based on a Lyapunov function, we show that all solutions $ (u, v) $ of ($\ast$) will exponentially converge to the unique constant steady state $ (1, 1) $ provided $ \mu>\frac{K_0}{16} $ with $ K_0 = \max\limits_{0\leq v \leq \infty}\frac{|\chi(v)|^2}{\gamma(v)} $.