Perturbing the boundary conditions of the generator of a cosine family
type: Item
instance of: scholarly article
title: Perturbing the boundary conditions of the generator of a cosine family
zbMATH Open document ID: 1262.47065
author: Edgardo Alvarez-Pardo
published in: Semigroup Forum
publication date: 2012-11-08T00:00:00Z
review text: The author considers the second order abstract Cauchy problem \[ \begin{gathered} u'' (t) = A_{\Phi} u (t), \;t \in \mathbb R, \\ u (0) = u_0, \;u' (0) = u_1, \end{gathered} \] where the operator \(A _{\Phi }\) is given by \(D(A _{\Phi }):=\{u\in D(A):Lu=\Phi u\}\) and \(A _{\Phi } u=Au\). Here, the operator \((A, D (A))\) is a given closed and densely defined linear operator defined on a Banach space \(X\), whereas \(L:D(A)\rightarrow \partial X\) and \({\Phi }:D(A)\rightarrow \partial X\) are closed linear operators and \(\partial X\) is another Banach space called boundary space. Imposing conditions on \(L\) and \(\Phi\), the author shows that the above second order abstract Cauchy problem is well-posed and thus generates a cosine operator function on the Banach space \(X\). Some examples are given.
Mathematics Subject Classification ID: 47D09
Mathematics Subject Classification ID: 34G10
zbMATH DE Number: 6103710
zbMATH Keywords: cosine functions
zbMATH Keywords: abstract Cauchy problems
zbMATH Keywords: general boundary conditions
reviewed by: Miklavž Mastinšek
MaRDI profile type: Publication
full work available at URL: s00233-011-9361-3
OpenAlex ID: W1978116950
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DOI: 10.1007/S00233-011-9361-3
description: scientific article; zbMATH DE number 6103710