Well-posedness for a class of mixed problem of wave equations

This paper studies an initial-boundary value problem (IBV) of the wave equation, in which time derivative of second order appears in the boundary condition. This results in study of new Sturm–Liouville problem. The solution of IBV is then constructed in terms of the family of eigenfunctions of the S-L problem. Uniqueness and stability are proved via energy inequality method. Some extension is explained. A parabolic variant is also illustrated.