Wave equation with second-order non-standard dynamical boundary conditions

The paper deals with the well-posedness of the problem u_tt-Δu=0 in ℝxΩ, u_tt=k u_ν on ℝxΓ, u(0,x)= u_0(x), u_t(0,x)= v_0(x) in Ω, where u = u(t, x), t ∈ ℝ, x ∈ Ω, Δ = Δx denotes the Laplacian operator with respect to the space variable, Ω is a bounded regular (C^∞) open domain of ℝ^N (N ≥ 1), Γ = ∂Ω, ν is the outward normal to Ω, 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 Ω is a ball.