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=\Delta u+\nabla \cdot(S(u) \nabla v), \tau v_t=\Delta v-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 $0<S(u)\leqslant K(1+u)^{\beta}$ for $u \geqslant 0$ 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 $S(u) \geqslant k u^{\beta}$ for $u \geqslant 0$ 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 $M\geqslant M^{\star}$, 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}, \textbf{153} (2023).], in which the authors studied the system ($\star$) with the first equation replaced respectively by $u_t=\nabla \cdot ((1+u)^{-\alpha} \nabla u)+\nabla \cdot(u \nabla v)$ and $u_t=\nabla \cdot ((1+u)^{-\alpha} \nabla u)+\nabla \cdot(\frac{u}{v} \nabla v)$. 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 $\alpha>0$.