Decay of solutions of the wave equation and spectral properties of the Laplace operator in expanding domains
type: Item
instance of: scholarly article
title: Decay of solutions of the wave equation and spectral properties of the Laplace operator in expanding domains
zbMATH Open document ID: 0915.35062
DOI: 10.1007/BF02316158
author: Alexey Filinovskiy
published in: Mathematical Notes
publication date: 1999-03-09T00:00:00Z
review text: Let \(\Omega\subset \mathbb{R}^n\), \(n\geq 2\), be an unbounded domain with boundary \(\Gamma\in C^2\). Consider the problem \[ u_{tt}- \Delta u= 0,\quad t>0,\quad x\in\Omega,\quad u(0, x)= f(x),\quad u_t(0, x)= g(x),\quad u|_\Gamma= 0,\tag{1} \] where \(f(x)\in \mathring W^1_2(\Omega)\) and \(g(x)\in L_2(\Omega)\) are real-valued initial functions. We say that the energy \[ E_\Omega(t)= \int_\Omega (u^2_t+|\nabla u|^2)dx= E_\Omega(0),\quad t>0 \] scatters to infinity if for any bounded domain \(\Omega'\subset \Omega\) we have \(\lim_{t\to\infty} E_{\Omega'}(t)= 0\), where \[ E_{\Omega'}(t)= \int_{\Omega'} (u^2_t+|\nabla u|^2)dx. \] We are interested in the relation between the scattering of energy to infinity for the solution of problem (1) and the spectral properties of the associated elliptic problem.
Mathematics Subject Classification ID: 35L05
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35P05
zbMATH DE Number: 1228266
zbMATH Keywords: scattering of energy to infinity
MaRDI profile type: Publication
cites work: Solvability in the small of nonstationary problems for incompressible ideal and viscous fluids and the case of vanishing viscosity
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description: scientific article; zbMATH DE number 1228266