Location of eigenvalues for the wave equation with dissipative boundary conditions

Vesselin  Petkov

doi:10.3934/ipi.2016034

Article Contents

2016, Volume 10, Issue 4: 1111-1139. Doi: 10.3934/ipi.2016034

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Location of eigenvalues for the wave equation with dissipative boundary conditions

Vesselin Petkov^1,

1.
Université de Bordeaux, Institut de Mathématiques de Bordeaux, 351, Cours de la Libération, 33405 Talence

Received: June 2015

Revised: March 2016

Published: November 2016

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Abstract

We examine the location of the eigenvalues of the generator $G$ of a semi-group $V(t) = e^{tG},\: t \geq 0,$ related to the wave equation in an unbounded domain $\Omega \subset \mathbb{R}^d$ with dissipative boundary condition $\partial_{\nu}u - \gamma(x) \partial_t u = 0$ on $\Gamma = \partial \Omega.$ We study two cases: $(A): \: 0 < \gamma(x) < 1,\: \forall x \in \Gamma$ and $(B):\: 1 < \gamma(x), \: \forall x \in \Gamma.$ We prove that for every $0 < \epsilon \ll 1,$ the eigenvalues of $G$ in the case $(A)$ lie in the region $\Lambda_{\epsilon} = \{ z \in \mathbb{C}:\: |Re z | \leq C_{\epsilon} (|Im z|^{\frac{1}{2} + \epsilon} + 1), \: Re z < 0\},$ while in the case $(B)$ for every $0 < \epsilon \ll 1$ and every $N \in \mathbb{N}$ the eigenvalues lie in $\Lambda_{\epsilon} \cup {\mathcal R}_N,$ where ${\mathcal R}_N = \{z \in \mathbb{C}:\: |Im z| \leq C_N (|Re z| + 1)^{-N},\: Re z < 0\}.$

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Mathematics Subject Classification: Primary: 35P20; Secondary: 47A40, 35L05.

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