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 Ω³ with smooth boundary ∂Ω≠∅. The key tools to prove local well-posedness are the dispersive estimates obtained in {L,L1,L3} and the Strichartz estimates in {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 R³ (see {MG}) and in any bounded domains in R³ (see {BLP}). Moreover, the result of the Strichartz estimates in our setting is strong enough when combined with the arguments in {BLP,SS2} so that we can extend local to global well-posedness.