Location of Eigenvalues for the Wave Equation with Dissipative Boundary Conditions

We examine the location of the eigenvalues of the generator G of a semi-group V (t) = etG, t ≥ 0, related to the wave equation in an unbounded domain Ω ⊂ Rd with dissipative boundary condition ∂νu − γ(x)∂tu = 0 on Γ = ∂Ω. We study two cases: (A) : 0 < γ(x) < 1, ∀x ∈ Γ and (B) : 1 < γ(x), ∀x ∈ Γ. We prove that for every 0 < 1, the eigenvalues of G in the case (A) lie in the region Λ = {z ∈ C : |Re z| ≤ C (|Im z| 1 2+ +1), Re z < 0}, while in the case (B) for every 0 < 1 and every N ∈ N the eigenvalues lie in Λ ∪RN , where RN = {z ∈ C : |Im z| ≤ CN (|Re z|+ 1)−N , Re z < 0}.