Time periodic solution to a mechanochemical model in biological patterns

In this paper, we consider a mechanochemical model in biological patterns in \begin{document}$ \mathbb{R}^N $\end{document}, \begin{document}$ N\geq 5 $\end{document}. We first prove the existence of time periodic solution in \begin{document}$ BC(\mathbb{R}; L^{N,\infty}(\Omega)) $\end{document}. Then we obtain the existence, uniqueness and regularity of the mild solution of the problem. Finally, we prove that the mild solution can become strong solution in \begin{document}$ BC(\mathbb{R}; L^{N,\infty}(\Omega)) $\end{document}.