### On the initial value problem for the hyperbolic Keller-Segel equation in the Besov framework

In this talk, we first show by constructing a special initial data that the solution map for the one dimensional hyperbolic Keller-Segel equations (HKSE) starting from $u_0$ is discontinuous at $t=0$ in the metric of $B_{2,\infty}^{s}(\mathbb{R})$, $s>\frac{3}{2}$. Then, we establish the Hadamard local well-posedness result for the high dimensional HKSE in the larger Besov spaces $B_{p,1}^{1+\frac{d}{p}}(\mathbb{R}^d)$, $1\leq p <\infty$. Moreover, we investigate the inviscid limit of the Keller-Segel equations with small diffusivity $\epsilon\Delta u$ as $\epsilon\rightarrow 0$ in the same topology of Besov spaces as the initial data. Finally, we establish two kinds of blow-up criteria for strong solutions in Besov spaces by means of the Littlewood-Paley theory. This is a joint work with Shouming Zhou, Lei Zhang and Simin Zhang.