Boundedness and finite-time blow-up in a repulsion-consumption system with nonlinear chemotactic sensitivity

This paper investigates the repulsion-consumption system \begin{align}\tag{$\star$} \left{ \begin{array}{ll} u_t=Δu+\nabla \cdot(S(u) \nabla v), τv_t=Δv-u v, \end{array} \right. \end{align} under no-flux/Dirichlet conditions for $u$ and $v$ in a ball $B_R(0)\subset {\mathbb{R}}^n$. When $\tau =\{0,1\}$ and $0<S(u)⩽K(1+u{)}^{\beta }$ for $u⩾0$ with some $\beta \in (0,\frac{n+2}{2n})$ and $K>0$, we show that for any given radially symmetric initial data, the problem ($\star$) possesses a global bounded classical solution. Conversely, when $\tau =0$, $n=2$ and $S(u)⩾k{u}^{\beta }$ for $u⩾0$ with some $\beta >1$ and $k>0$, for any given initial data $u_0$, there exists a constant $M^{\star }=M^{\star }\left(u_0\right)>0$ with the property that whenever the boundary signal level $M⩾M^{\star }$, the corresponding radially symmetric solution blows up in finite time. Our results can be compared with that of the papers [J.~Ahn and M.~Winkler, {\it Calc. Var.} {\bf 64} (2023).] and [Y. Wang and M. Winkler, {\it Proc. Roy. Soc. Edinburgh Sect. A}, $\mathbf{\text{153}}$ (2023).], in which the authors studied the system ($\star$) with the first equation replaced respectively by $u_t=\mathrm{\nabla }\cdot ((1+u{)}^{-\alpha }\mathrm{\nabla }u)+\mathrm{\nabla }\cdot (u\mathrm{\nabla }v)$ and $u_t=\mathrm{\nabla }\cdot ((1+u{)}^{-\alpha }\mathrm{\nabla }u)+\mathrm{\nabla }\cdot (\frac{u}{v}\mathrm{\nabla }v)$. Among other things, they obtained that, under some conditions on $u_0(x)$ and the boundary signal level, there exists a classical solution blowing up in finite time whenever $\alpha >0$.