Stabilization of the solutions of an exterior Dirichlet boundary-value problem for the equation of the oscillations of a plate
type: Item
instance of: scholarly article
title: Stabilization of the solutions of an exterior Dirichlet boundary-value problem for the equation of the oscillations of a plate
zbMATH Open document ID: 0618.73069
DOI: 10.1007/BF01158005
published in: Mathematical Notes
publication date: 1986-01-01T00:00:00Z
review text: Let \(\Omega \subset E_ n\) \((n\geq 2)\) be an unbounded domain with compact boundary \(\Gamma\). In the semiinfinite cylinder \((t>0)\times \Omega\) the author considers the equation \[ \frac{\partial^ 2u} {\partial t^ 2}+ (-\Delta)^ pu=0,\qquad p\geq 2, \tag{1} \] with the initial conditions \[ u|_{t=0}=f_ 0(x),\qquad \frac{\partial u}{\partial t}|_{t=0}=f_ 1(x), \tag{2} \] and the boundary conditions \[ \frac{\partial^ ju}{\partial n^ j_ x}|_{x\in \Gamma}=0,\qquad j=0,...,p-1 \text{ if } t>0, \tag{3} \] where \(n_ x\) is normal to the surface \(\Gamma\). If the surface \(\Gamma\) is sufficiently smooth and the functions \(f_ i(x),i=0,1\), are smooth and finite in \(\Omega\) then the author shows that the solution of the exterior Dirichlet boundary-value problem (1)-(3) tends towards zero as \(t\to \infty\), uniformly on each compactum of \({\bar \Omega}\).
Mathematics Subject Classification ID: 74H45
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 74K20
zbMATH DE Number: 4003654
zbMATH Keywords: unbounded domain
zbMATH Keywords: compact boundary
zbMATH Keywords: semiinfinite cylinder
zbMATH Keywords: exterior Dirichlet boundary-value problem
author: Alexey Filinovskiy
reviewed by: G. Jaiani
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4003654