Global boundedness in a quasilinear chemotaxis–consumption system with degenerate signal-dependent motility and logistic source

In this paper, we deal with the following quasilinear chemotaxis–consumption system with degenerate signal-dependent motility \(\begin{aligned} \left\{ \begin{array}{lll} u_t=\nabla \cdot (v^{\alpha }\nabla u^{m})+\alpha \nabla \cdot (v^{\alpha -1}u\nabla v)+ru-\mu u^{2},\quad & x\in \Omega ,~t>0,\\ v_t=\Delta v-uv, & x\in \Omega ,~t>0,\\ \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{u}_t=\mathrm{\nabla }·(v^{\alpha }\mathrm{\nabla }{u}^m)+\alpha \mathrm{\nabla }·(v^{\alpha -1}u\mathrm{\nabla }v)+ru-\mu u^2,\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ v_t=\mathrm{\Delta }v-uv,\hfill & x\in \mathrm{\Omega },t>0,\hfill \end{array}\right)\end{array}$$ under the homogeneous Neumann boundary condition in \(\Omega \subset \mathbb {R}^{n}(n\ge 1)\) $\mathrm{\Omega }\subset {\mathbb{R}}^n(n\ge 1)$ with \(\alpha ,~r,~\mu >0\) $\alpha ,r,\mu >0$ and \(m>1\) $m>1$ . It is shown that above system admits at least one global weak solution fulfilling the following boundedness property \(\begin{aligned} \Vert u(\cdot ,t)\Vert _{L^{p}(\Omega )}+\Vert v(\cdot ,t)\Vert _{W^{1,\infty }(\Omega )}\le C \end{aligned}\) $$\begin{array}{c}\hfill {\Vert u(·,t)\Vert }_{L^p(\mathrm{\Omega })}+{\Vert v(·,t)\Vert }_{W^{1,\infty }(\mathrm{\Omega })}\le C\end{array}$$ for all \(t>0\) $t>0$ ,  \(p>2\) $p>2$ and \(m>1\) $m>1$ . This result gives an evident of the regularized effect of porous-medium-type diffusion.