Time periodic solution to a mechanochemical model in biological patterns

Abstract

<p style='text-indent:20px;'>In this paper, we consider a mechanochemical model in biological patterns in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ N\geq 5 $\end{document}</tex-math></inline-formula>. We first prove the existence of time periodic solution in <inline-formula><tex-math id="M3">\begin{document}$ BC(\mathbb{R}; L^{N,\infty}(\Omega)) $\end{document}</tex-math></inline-formula>. Then we obtain the existence, uniqueness and regularity of the mild solution of the problem. Finally, we prove that the mild solution can become strong solution in <inline-formula><tex-math id="M4">\begin{document}$ BC(\mathbb{R}; L^{N,\infty}(\Omega)) $\end{document}</tex-math></inline-formula>.</p>