Global classical solutions to a chemotaxis consumption model involving singularly signal-dependent motility and logistic source

This work considers the Keller–Segel consumption system $\left(\begin{array}{cc}{u}_t=\Delta \left(u{v}^{-\alpha }\right)+au-b{u}^{\gamma },& x\in \Omega ,t>0,\\ v_t=\Delta v-uv,& x\in \Omega ,t>0\end{array}\right)$under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega \subset R^n,n\ge 2$, where the parameters $a>0$, $b>0$, $\gamma \ge 2$ and $\alpha \in (0,1)$, the initial data $u_0\in C^0\left(\bar{\Omega }\right)$, $v_0\in W^{1,\infty }\left(\Omega \right)$, $u_0\ge 0(⁄\equiv 0)$ and $v_0>0$ in $\bar{\Omega }$ with ${\Vert v_0\Vert }_{L^{\infty }\left(\Omega \right)}<exp\left(\frac{ln\left(\frac{1-\alpha }{\alpha }\cdot \frac{8}{n}\right)}{\alpha }\right).$It is shown that if one of the following cases holds:

(i) $\gamma >2$;

(ii) $\gamma =2$ and $b>\frac{(n-2)\alpha }{n}$,

then the corresponding initial–boundary value problem possesses global classical solutions.