2015/04/24 by Vesselin Petkov, Petkov, Vesselin
Mathematics · Physics and Astronomy · #35L05 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Primary 35P20 #Secondary 47A40 #math-ph #math.AP #math.MP #msc:35L05 #msc:35P20 #msc:47A40
paper · pdf · doi:10.48550/arxiv.1504.06408
Some proofs in the Appendix of the version 3 are corrected
arxiv created 2016/03/24 · arxiv updated 2016/03/25
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⊂ \mathbb Rd with dissipative boundary condition ∂νu - γ(x) ∂t u = 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 < ε≪ 1, the eigenvalues of G in the case (A) lie in the region Λε = \z ∈ \mathbb C: |\Re z | ≤ Cε (|\Im z|(1)/(2) + ε + 1), \Re z < 0\, while in the case (B) for every 0 < ε≪ 1 and every N ∈ \mathbb N the eigenvalues lie in Λε ∪ \mathcal RN, where \mathcal RN = \z ∈ \mathbb C: |\Im z| ≤ CN (|\Re z| + 1)-N, \Re z < 0\.