Spectrum and stabilization in hyperbolic problems
type: Item
instance of: scholarly article
title: Spectrum and stabilization in hyperbolic problems
zbMATH Open document ID: 1409.35132
published in: Journal of Mathematical Sciences (New York)
publication date: 2018-11-02T00:00:00Z
review text: The authors studies the mixed problem \[ \begin{gathered} u_{tt}-\Delta u=0,\quad t>0, \quad x\in\Omega, \\ u=0,\quad x\in\partial\Omega, \qquad u(0,x)=f(x),\qquad u_t(0,x)=g(x). \end{gathered} \] Here \(\Omega\subseteq\mathbb{R}^n\), \(n\geqslant 2\), which can be also unbounded. The main aim is to study the relationship between spectral properties of the Laplace operator and the stabilization of the solutions as \(t\to+\infty\). Denote \(\mathcal{E}_{\Omega'}(t):=\|\nabla u\|_{L_2(\Omega')}^2\). The main results are as follows. The first result states that if \[ \lim_{t\to+\infty} \frac{1}{t}\int_{0}^{t}\mathcal{E}_{\Omega'(\tau)}d\tau=0 \tag{1} \] for each \(\Omega'\subset\Omega\) and \(\Omega\) is bounded, then \(u=0\). The second result concerns an unbounded domain \(\Omega\) and it says that, if the point spectrum of the Dirichlet Laplacian in \(\Omega\) is non-empty, then there exist functions \(f, g \in C_0^\infty(\Omega)\) and a bounded domain \(\Omega'\subset\Omega\) such that \[ \liminf_{t\to+\infty} \mathcal{E}_{\Omega'}(t)>0. \] If the point spectrum is empty, then for \(f\in \overset{\circ}{H}^1(\Omega)\), \(g\in L_2(\Omega)\) and bounded domains \(\Omega'\subset\Omega\) we again have (1). If in addition we know that the spectrum of the Dirichlet Laplacian is absolutely continuous, then \[ \lim_{t\to+\infty} \mathcal{E}_{\Omega'}(t)=0. \] The paper also contains a series of theorems providing more detailed information about the decay of \(\mathcal{E}_{\Omega'}(t)\).
Mathematics Subject Classification ID: 35L20
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35L05
zbMATH DE Number: 6971113
zbMATH Keywords: Dirichlet Laplacian
zbMATH Keywords: behavior of solution at infinity
author: Alexey Filinovskiy
reviewed by: Denis Borisov
MaRDI profile type: Publication
full work available at URL: s10958-018-4027-2
OpenAlex ID: W2889242841
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Wikidata QID: Q129333866
DOI: 10.1007/S10958-018-4027-2
description: scientific article; zbMATH DE number 6971113