Boundedness of classical solutions to a chemotaxis consumption model with signal-dependent motility

This paper deals with the following chemotaxis system: \(\begin{aligned} \left\{ \begin{array}{ll} u_{t}=\nabla \cdot \big (\gamma (v) \nabla u-u \,\xi (v) \nabla v\big )+\mu \, u\,(1-u), &{} x\in \Omega , \ t>0, \\ v_{t}=\Delta v-uv, &{} x\in \Omega , \ t>0, \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{u}_t=\mathrm{\nabla }·(\gamma (v)\mathrm{\nabla }u-u\xi (v)\mathrm{\nabla }v)+\mu u(1-u),\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ v_t=\mathrm{\Delta }v-uv,\hfill & x\in \mathrm{\Omega },t>0,\hfill \end{array}\right)\end{array}$$ under homogeneous Neumann boundary conditions in a bounded domain \( \Omega \subset {\mathbb {R}}^{n}, n\ge 2,\) $\mathrm{\Omega }\subset {\mathbb{R}}^n,n\ge 2,$ with smooth boundary. Here, the positive function \(\gamma \in C ^{2}([0, +\infty )) \) $\gamma \in C^2([0,+\infty ))$ satisfies \(\gamma '(s)<0\) ${\gamma }^{\prime }(s)<0$ and \( \gamma ''(s)\ge 0\) ${\gamma }^{\prime \prime }(s)\ge 0$ for all \(s\ge 0,\) $s\ge 0,$ also \(\xi (s)= -(1-\alpha )\,\gamma '(s) \) $\xi (s)=-(1-\alpha ){\gamma }^{\prime }(s)$ with \(\alpha \in (0, 1)\) $\alpha \in (0,1)$ . For the above system, we prove that the corresponding initial boundary value problem admits a unique global classical solution which is uniformly in time bounded. This result is obtained for small initial data without any restriction on \(\mu .\) $\mu .$ The obtained result improves a recent result by Li and Lu (J Math Anal Appl 521:126902, 2023), which asserts the global existence of bounded classical solutions, provided that \( \frac{(\gamma '(s))^{2}}{\gamma ''(s)} \le \frac{n}{2(n+1)^{3}}\) $\frac{{({\gamma }^{\prime }(s))}^2}{{\gamma }^{\prime \prime }(s)}\le \frac{n}{2{(n+1)}^3}$ and some conditions on initial data and \(\mu .\) $\mu .$ We should mention that in the special cases \(\gamma (s)=(1+s)^{-k}\,(k>0)\) $\gamma (s)={(1+s)}^{-k}(k>0)$ and \(\gamma (s)=\textit{e}^{-\chi s}\, (\chi >0),\) $\gamma (s)={\mathit{e}}^{-\chi s}(\chi >0),$ the result in Li and Lu (2023) is obtained under conditions on k and \(\chi .\) $\chi .$ But, our result is without any restriction on k and \(\chi .\) $\chi .$