On a system of Klein-Gordon type equations with acoustic boundary conditions
type: Item
instance of: scholarly article
title: On a system of Klein-Gordon type equations with acoustic boundary conditions
zbMATH Open document ID: 1060.35118
author: A. T. Cousin
author: Nikolai A. Larkin
author: C. L. Frota
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-08-06T00:00:00Z
review text: This paper deals with the existence, uniqueness and asymptotic behaviour of solutions to the \((k\times k)\) system of Klein-Gordon type equations \[ \begin{aligned} u_1^{\prime\prime} &- \Delta_x u_1+ \alpha_1 u_1+ a_{12} u_1 u^2_2+ a_{13} u_1u^2_3+\cdots+ a_{1k} u_1 u^2_k= f_1,\\ u^{\prime\prime}_k&- \Delta_x u_k+ \alpha_k u_k+ a_{k1} u_k u^2_1+ a_{k2} u_k u^2_2+\cdots+ a_{k(k-1)} u_k u^2_{k-1}= f_k, \end{aligned}\tag{1} \] where \(\alpha_i\) \((1\leq i\leq k)\), \(a_{ij}= a_{ji}\) are nonnegative constants, \(a_{ii}= 0\). The authors prove the existence of global weak solutions to (1) and the exponential decay of the energy for \(f_i\equiv 0\) and suitable boundary and initial conditions.
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 81Q05
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 2082145
zbMATH Keywords: exponential decay
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: asymptotic behaviour
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.01.007
OpenAlex ID: W2054477583
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DOI: 10.1016/J.JMAA.2004.01.007
description: scientific article; zbMATH DE number 2082145