This paper deals with the following chemotaxis system with signal-dependent motility and logistic source \(\begin{aligned} \left\{ \begin{array}{llll} u_t=\nabla \cdot (\gamma (v)\nabla u-u\xi (v)\nabla v)+\mu u(1-u),\quad & x\in \Omega ,\quad t>0,\\ v_t=\Delta v-uv,\quad & x\in \Omega ,\quad t>0\\ \end{array} \right. \end{aligned}\) under homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega \subset {\mathbb {R}}^n\) ( \(n\ge 1\) ). Here the parameter \(\mu >0\) , the function \(\gamma \in C^2([0,\infty ))\) satisfies \(\gamma (s)>0\) for all \(s\ge 0\) , and \(\xi (s)=-(1-\alpha )\gamma '(s)\) with \(\alpha \in (0,1)\) . For all suitably regular initial data, if one of the following cases holds: (i) \(n\le 2\) ;
(ii) \(n\ge 3\) and
\(\begin{aligned} \mu > \frac{n(n-2)}{2n+4}\left( \frac{4(n^2+n)}{n+2}\right) ^\frac{2}{n}\Vert M v_0\Vert _{L^\infty (\Omega )}^{\frac{4}{n}}+\frac{16(n-1)}{n+2}\left( \frac{16n(n^2-1)}{n+2}\right) ^\frac{n-2}{4}\Vert Mv_0\Vert _{L^\infty (\Omega )}^{n} \end{aligned}\) with \(M:=\frac{(1-\alpha )\max \limits _{0 \le s \le \Vert v_0\Vert _{L^\infty (\Omega )} }|\gamma '(s)|}{\sqrt{\min \limits _{0 \le s \le \Vert v_0\Vert _{L^\infty (\Omega )} }\gamma (s)}}\) , then this system possesses global classical solutions which are uniformly bounded, while if \(n\ge 3\) and \(\mu >0\) , this system admits at least one global weak solution. This work improves the results in [1, 2, 16].
