The role of logistic damping in a class of chemotaxis systems with density-dependent motility and indirect signal absorption

This paper deals with the chemotaxis consumption model with signal-dependent motility and indirect signal absorption$ \begin{equation*} \begin{cases} u_t = \nabla\cdot(D(v)\nabla u-uS(v)\nabla v)+\rho u-\mu u^l,&x\in\Omega,\,t>0,\\ v_t = \Delta v-vw,&x\in\Omega,\,t>0,\\ w_t = -\delta w+u,&x\in\Omega,\,t>0,\\ \end{cases} \end{equation*} $under homogeneous Neumann boundary conditions. The domain $ \Omega\subset\mathbb{R}^n\ (n\geqslant1) $ has a smooth boundary. The initial data $ u_0\in W^{1,\infty}(\Omega) $, $ v_0\in W^{1,\infty}(\Omega) $ and $ w_0\in C^{\beta}(\overline{\Omega}) $ are nonnegative. The function $ D $ satisfies $ D\in C^3([0,+\infty)) $, $ D>0\,\text{on}\,[0,+\infty) $ and $ S $ satisfies $ S\in C^3([0,+\infty)) $. The parameters $ \rho>0 $ and $ \delta>0 $. Then we establish the global boundedness existence of solutions for such kind of models under the condition that $ \mu $ is large enough if $ l = 2(n\geqslant4) $ and $ \mu>0 $ if $ l>\min\left\{\max\{\frac{n}{2},1\},2\right\} $. Furthermore, using Lyapunov functionals, it is shown the global bounded solution will converge to $ \left((\frac \rho \mu)^{\frac{1}{l-1}},0,\frac{1}{\delta}(\frac \rho \mu)^{\frac{1}{l-1}}\right) $ as $ t\rightarrow +\infty $. More precisely, if $ l\geqslant2 $ the convergence rate is exponential and if $ 1<l<2 $ it is algebraic.