LOCATION OF EIGENVALUES FOR THE WAVE EQUATION WITH DISSIPATIVE BOUNDARY CONDITIONS

Vesselin Petkov (1)

1 IMB - Institut de Mathématiques de Bordeaux

We examine the location of the eigenvalues of the generator G of a semi-group V (t) = e tG , t ≥ 0, related to the wave equation in an unbounded domain Ω ⊂ R d with dissipative boundary condition ∂ν u − 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 ∪ R N , where R N = {z ∈ C : | Im z| ≤ C N (| Re z| + 1) −N , Re z < 0}.

Mots clés: dissipative boundary conditions; location of the spectrum of a generator; asymptotically disapearing solitions

Domaines: Equations aux dérivées partielles [math.AP]; Théorie spectrale [math.SP]

Vesselin Petkov. LOCATION OF EIGENVALUES FOR THE WAVE EQUATION WITH DISSIPATIVE BOUNDARY CONDITIONS. 2016. ⟨hal-01256216v1⟩