Polynomial decay to a class of abstract coupled system with past history.
type: Item
instance of: scholarly article
title: Polynomial decay to a class of abstract coupled system with past history.
zbMATH Open document ID: 1274.35024
author: L. P. V. Matos
author: D. S. Almeida Júnior
author: Mauro Santos
published in: Differential and Integral Equations
publication date: 2013-03-21T00:00:00Z
review text: The authors study an initial value problem for a class of coupled systems of the form \[ u_{tt}+ A_1 u-\int _0^{\infty }g(s)A_2 u(t-s) ds+\beta v=0, \] \[ v_{tt} +Bv+\beta u=0, \] where \(A_i, B\) are self-adjoint operators on a Hilbert space satisfying \(A_2=f(A_1),\;B=h(A_1)\) with \(f(s)=o(s^{\alpha }),\;h(s)=o(s^{\gamma })\) as \(s\to \infty \). The function \(g\) is assumed to be a nonnegative integrable function satisfying \(g'(s)\leq a g(s)\), \(s\in \mathbb {R}^{+}\). Using the relative history of \(u\), the authors rewrite the system in the abstract form \[ \frac {\text{d}}{\text{d}t}U(t)=\mathcal {A} U(t), \] and show that the semigroup associated with the system is not exponentially stable when \(\gamma >0\) and \(\alpha <1.\) It is shown that solutions decay polynomially to zero in an appropriate norm, with rates that can be improved by taking more regular initial data. Some examples conclude the paper.
reviewed by: Hana Petzeltová
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35L70
zbMATH DE Number: 6148069
zbMATH Keywords: coupled systems with past history
zbMATH Keywords: polynomial decay of solutions
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6148069