| Abstract: |
| In this presentation, we will discuss recent progress for a repulsion-consumption system with nonlinear chemotactic sensitivity. The consumption system differs from Keller-Segel production systems, and the literature on solutions for consumption systems primarily focuses on global existence. Inspired by [J. Ahn and M. Winkler, Calc. Var. 64 (2023).] and [Y. Wang and M. Winkler, Proc. Roy. Soc. Edinburgh Sect. A, 153 (2023).], we investigate the effect of the nonlinear chemotactic sensitivity $S(u)=(1+u)^\beta$ on the occurrence of blow-up phenomenon for the repulsion-consumption parabolic-elliptic system and establish the boundedness of solutions for the repulsion-consumption system to find the critical exponent. Under radially symmetric assumptions, we prove that
1) The signal consumption equation is elliptic and $n=2$. For $\beta>1$ and the boundary signal level large enough, the corresponding radially symmetric solution blows up in finite time.
2) When the signal consumption equation is elliptic or parabolic and $n \geq 2$. For $\beta \in\left(0, \frac{n+2}{2 n}\right)$ the problem $(\star)$ possesses a global bounded classical solution. |
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