Global well-posedness for a quasilinear chemotaxis system with decaying diffusivity and consumption of a chemoattractant

The Neumann initial-boundary value problem for the quasilinear parabolic-parabolic chemotaxis model:$\{\begin{array}{cc}{\partial }_{t}u=\nabla \cdot \left(D\right(u\left)\nabla u\right)-\nabla \cdot \left(S\right(u\left)\nabla v\right),& x\in \Omega ,t>0,\\ {\partial }_{t}v=\Delta v-uv,& x\in \Omega ,t>0,\end{array}$ is considered in smoothly bounded domains $\Omega \subset R^n$, $n\ge 1$, where $D\in C^2\left(\right[0,\infty \left)\right)$ and $S\in C^2\left(\right[0,\infty \left)\right)$. The diffusivity sensitivity $D\left(s\right)$ and chemotaxis sensitivity $S\left(s\right)$ satisfy$K_1{e}^{-{\beta }^{-}s}\le D\left(s\right)\le K_2{e}^{-{\beta }^{+}s}\text{and}\frac{S\left(s\right)}{D\left(s\right)}\le K_3{e}^{\alpha s},\text{ for any}s\ge 0,$ $S\left(0\right)=0<S\left(s\right)$ for all $s>0$, and $K_i>0$ $(i=1,2,3)$, ${\beta }^{-}>0$, ${\beta }^{+}\in (-\infty ,{\beta }^{-}]$. We show that the global existence of classical solution as $\alpha \in (-\infty ,\frac{{\beta }^{+}}{2})$, and the classical solution are globally bounded if Ω is convex, ${\beta }^{-}\ge {\beta }^{+}>0$ and $\alpha \in (-\infty ,\frac{{\beta }^{+}}{n+1})$.