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

This paper investigates the repulsion-consumption system {align}{} \{ {array}{ll} u_t=Δ u+∇ ·(S(u) ∇ v), τ v_t=Δ v-u v, {array} . {align} under no-flux/Dirichlet conditions for u and v in a ball BR(0) ⊂ Rⁿ. When τ=\0,1\ and 0<S(u) K(1+u)β for u 0 with some β ∈ (0,n+2/2n) and $K>0$, we show that for any given radially symmetric initial data, the problem () possesses a global bounded classical solution. Conversely, when τ=0, $n=2$ and S(u) k uβ for u 0 with some β>1 and $k>0$, for any given initial data u₀, there exists a constant M^=M^(u₀)>0 with the property that whenever the boundary signal level M M^, 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, { Calc. Var.} { 64} (2023).] and [Y. Wang and M. Winkler, { Proc. Roy. Soc. Edinburgh Sect. A}, 153 (2023).], in which the authors studied the system () with the first equation replaced respectively by uₜ=∇ · ((1+u)-α ∇ u)+∇ ·(u ∇ v) and uₜ=∇ · ((1+u)-α ∇ u)+∇ ·(u/v ∇ v). Among other things, they obtained that, under some conditions on u₀(x) and the boundary signal level, there exists a classical solution blowing up in finite time whenever α>0.