Lecture Notes ( Wuhan 2006 )

By linearity, the above two questions are reduced to the unique continuation property (UCP) for the wave equation. Due to finite speed of propagation, the (UCP) holds only for T > 0 large enough and Ω will be supposed a connected domain. For instance, for a sufficiently smooth boundary, if T > 2max { dist (x, Γ) , x ∈ Ω}, then by Holmgren uniqueness theorem N = {u ∈ H (Ω× (0, T )) being a solution of (1.1) such that ∂nu = 0 on Γ× (0, T ) } = {0} .