Effects of indirect signal absorption in the chemotaxis system involving singularly signal‐dependent motilities

We consider the initial‐boundary problem of the chemotaxis system with indirect consumption ut=Δ(uv−α),x∈Ω,t>0,vt=Δv−vz,x∈Ω,t>0,zt=Δz−z+u,x∈Ω,t>0$$\begin{equation*} {\hspace*{92.5pt}\begin{cases} u_t=\Delta (uv^{-\alpha }),\ \ \ &x\in \Omega,\ t>0,\\[4pt] v_t=\Delta v-vz,\ \ \ &x\in \Omega,\ t>0,\\[4pt] z_t=\Delta z-z+u,\ \ \ &x\in \Omega,\ t>0 \end{cases}} \end{equation*}$$in a smoothly bounded domain Ω⊂Rn$\Omega \subset \mathbb {R}^n$ with n∈{2,3}$n\in \lbrace 2,3\rbrace$. It is shown that for all suitably regular initial data, global solvability in classical sense can be established even with singularly signal‐dependent motilities involved. To be specific, the global classical solution is constructed for arbitrary α>0$\alpha >0$.