Location of eigenvalues for the wave equation with dissipative boundary conditions

TL;DR

This paper analyzes the spectral location of eigenvalues of the wave equation generator with dissipative boundary conditions, revealing different eigenvalue regions depending on the boundary dissipation parameter.

Contribution

It provides precise eigenvalue location regions for the wave equation with dissipative boundary conditions, distinguishing cases based on the boundary dissipation parameter (x).

Findings

Eigenvalues in case (A) lie in a specific region _psilon.

Eigenvalues in case (B) are confined to _psilon and a rapidly decaying region _N.

Results depend on the boundary dissipation parameter (x).

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 ${\mathcal R}_N = \{z \in {\mathbb C}:\: |\Im z| \leq C_N (|\Re z| + 1)^{-N},\: \Re z < 0\}.$