Global well-posedness, blow-up phenomenon and ill-posedness for the hyperbolic Keller-Segel equations

In this paper, we consider the Cauchy problem of the hyperbolic Keller-Segel equations in $H^s\left(T^d\right)$ on torus with $d\ge 1$. Firstly, developing the dissipative mechanism through translation, we establish the global well-posedness in $H^s\left(T^d\right)$ ($s>1+\frac{d}{2}$) with initial data near some equilibrium state. Secondly, by capturing the feature of the preservation of zero directional derivative, we give a class of initial date that lead to finite time blow-up. It's worth noting that our method of proving blow-up phenomenon does not require any conservation law. Finally, the characterization of this blow-up motivates us to show the ill-posedness of this system in $H^{\frac{3}{2}}\left(T^d\right)$ in the sense of “norm inflation”, which implies that our ill-posedness result for this system is sharp on one dimensional torus.