Boundedness of classical solutions to a chemotaxis consumption system with signal dependent motility and logistic source

We consider the chemotaxis system: \begin{equation*} {\left\lbrace \begin{array}{ll} u_{t}=\nabla \cdot \big (\gamma (v) \nabla u-u \,\xi (v) \nabla v\big )+\mu \, u(1-u), & x\in \Omega , \ t>0, \\ v_{t}=\Delta v-uv, & x\in \Omega , \ t>0, \end{array}\right.} \end{equation*} under homogeneous Neumann boundary conditions in a bounded domain $ \Omega \subset \mathbb{R}^{n}, n\ge 2,$ with smooth boundary. Here, the functions $\gamma (v)$ and $\xi (v)$ are as: \begin{equation*} \gamma (v)=(1+v)^{-k}\quad \text{and} \quad \xi (v)=-(1-\alpha )\gamma ^{\prime }(v), \end{equation*} where $k>0$ and $\alpha \in (0,1).$We prove that the classical solutions to the above system are uniformly-in-time bounded provided that $ k\,(1-\alpha ) \frac{k\,n\,(1-\alpha ) \Vert v_{0}\Vert _{L^{\infty }(\Omega )}}{(n+1)(1+\Vert v_{0}\Vert _{L^{\infty }(\Omega )})}. \end{equation*} This result improves the recent result obtained for this problem by Li and Lu (J. Math. Anal. Appl.) (2023).