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 uₜ= u+ (S (u) v), vₜ= v-u v, array. align under no-flux/Dirichlet conditions for u and v in a ball BR (0) Rⁿ. When =\0, 1\ and 00, we show that for any given radially symmetric initial data, the problem () possesses a global bounded classical solution. Conversely, when =0, n=2 and S (u) k u^ for u 0 with some >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) ^- u) + (u v) and uₜ= ( (1+u) ^- u) + (uv 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 >0.