WAVE EQUATION WITH SECOND-ORDER NON-STANDARD DYNAMICAL BOUNDARY CONDITIONS

The paper deals with the well-posedness of the problem {u(tt) - Delta u = 0 in R x Omega, u(tt) = ku(v) on R x Gamma, u(0,x) = u(0)(x), u(t)(0,x) = v(0)(x) in Omega, where u = u(t, x), t is an element of R, x is an element of Omega, Delta = Delta(x) denotes the Laplacian operator with respect to the space variable, Omega is a bounded regular (C-infinity) open domain of R-N (N >= 1), Gamma = partial derivative Omega, v is the outward normal to Omega, k is a constant. We prove that it is ill-posed if N >= 2, while it is well-posed when N = 1. In the one-dimensional case, we give a complete existence, uniqueness and regularity theory. We also give some existence result for regular initial data when N >= 2 and Omega is a ball.