Well-Posedness for Quintic Energy Critical Wave in 3D Cylindrical Convex Domains

In this paper, we establish the well-posedness in energy space for the quintic energy critical wave inside a cylindrical convex domain $\mathrm{\Omega }\subset {\mathbb{R}}^3$ with smooth boundary $\mathrm{\partial }\mathrm{\Omega }\mathrm{\ne }\mathrm{\varnothing }$. The key tools to prove local well-posedness are the dispersive estimates obtained in \cite{L,L1,L3} and the Strichartz estimates in \cite{L2}. We point out that our result on the local and global existence of the solution to the wave equation in the cylindrical domain setting interpolates between that of in Euclidean space ${\mathbb{R}}^3$ (see \cite{MG}) and in any bounded domains in ${\mathbb{R}}^3$ (see \cite{BLP}). Moreover, the result of the Strichartz estimates in our setting is strong enough when combined with the arguments in \cite{BLP,SS2} so that we can extend local to global well-posedness.