Approaching logarithmic singularities in quasilinear chemotaxis-consumption systems with signal-dependent sensitivities

The chemotaxis system\begin{document}$ \begin{array}{l}\left\{ \begin{array}{l} u_t = \nabla \cdot \big( D(u) \nabla u \big) - \nabla \cdot \big( uS(x, u, v)\cdot \nabla v\big), \\ v_t = \Delta v -uv, \end{array} \right. \end{array} $\end{document}is considered in a bounded domain \begin{document}$ \Omega\subset \mathbb{R}^n $\end{document} , \begin{document}$ n\ge 2 $\end{document} , with smooth boundary. It is shown that if \begin{document}$ D: [0, \infty) \to [0, \infty) $\end{document} and \begin{document}$ S: \overline{\Omega}\times [0, \infty)\times (0, \infty)\to \mathbb{R}^{n\times n} $\end{document} are suitably smooth functions satisfying\begin{document}$ \begin{array}{l}D(u) \ge k_D u^{m-1} \qquad {\rm{for\; all}}\; u\ge 0 \end{array} $\end{document}and\begin{document}$ \begin{array}{l}|S(x, u, v)| \le \frac{S_0(v)}{v^\alpha} \qquad {\rm{for\; all}}\; (x, u, v)\; \in \Omega\times (0, \infty)^2 \end{array} $\end{document}with some\begin{document}$ \begin{array}{l}m>\frac{3n-2}{2n} \qquad {\rm{and}}\;\alpha\in [0, 1), \end{array} $\end{document}and with some \begin{document}$ k_D>0 $\end{document} and nondecreasing \begin{document}$ S_0: (0, \infty)\to (0, \infty) $\end{document} , then for all suitably regular initial data a corresponding no-flux type initial-boundary value problem admits a global bounded weak solution which actually is smooth and classical if \begin{document}$ D(0)>0 $\end{document} .