Global existence and optimal decay rate of solutions to hyperbolic chemotaxis system in Besov spaces

In this paper, we study the qualitative behavior of hyperbolic system arising from chemotaxis models. Firstly, by establishing a new product estimates in multi-dimensional Besov space \begin{document}$ \dot{B}_{2, r}^{\frac d2}(\mathbb{R}^d)(1\leq r\leq \infty) $\end{document} , we establish the global small solutions in multi-dimensional Besov space \begin{document}$ \dot{B}_{2, r}^{\frac d2-1}(\mathbb{R}^d) $\end{document} by the method of energy estimates. Then we study the asymptotic behavior and obtain the optimal decay rate of the global solutions if the initial data are small in \begin{document}$ B_{2, 1}^{\frac{d}{2}-1}(\mathbb{R}^d)\cap \dot{B}_{1, \infty}^0(\mathbb{R}^d) $\end{document} .