Energy decay for a damped nonlinear hyperbolic equation
type: Item
instance of: scholarly article
title: Energy decay for a damped nonlinear hyperbolic equation
zbMATH Open document ID: 0942.35118
DOI: 10.1016/S0893-9659(98)00171-2
published in: Applied Mathematics Letters
publication date: 2000-08-23T00:00:00Z
review text: This paper concentrates on the energy decay of the solution to the initial boundary value problem for a damped nonlinear hyperbolic equation \[ u''+ k_1\Delta^2u+ k_2\Delta^2 u'+\Delta g(\Delta u)= 0\quad\text{in }\Omega\times \mathbb{R}^+,\tag{1} \] \[ u= 0,\quad{\partial u\over\partial\nu}= 0\quad\text{on }\Gamma\times \mathbb{R}^+,\tag{2} \] \[ u(x,0)= u_0(x),\quad u'(x,0)= u_1(x)\quad\text{on }\Omega,\tag{3} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) with smooth boundary \(\Gamma\), \(k_1\) and \(k_2\) are two positive constants, \(g\) is a \(C^2\) class real-valued function, \(u_0\) and \(u_1\) are given functions on \(\Omega\). The global existence of weak solutions by using a variational method and the semigroup formulation have been proven, so the present note deals with the decay properties of solutions of equations (1), (2), (3) and establishes the uniform stability of solutions. The main theorem of the paper states that under certain conditions the relation \(E(t)\leq E(0)\exp(1-\omega t)\) for all \(t\in\mathbb{R}^+\) holds, where \[ E(t)={1\over 2} \int_\Omega (|u'|^2+ k_1|\Delta u|^2) dx+ \int_\Omega G(\Delta u) dx,\quad G(x)= \int^x_0 g(t) dt. \]
Mathematics Subject Classification ID: 35L75
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35L35
zbMATH DE Number: 1369372
zbMATH Keywords: uniform stabilization
zbMATH Keywords: strong damping
zbMATH Keywords: integral inequality
author: Mohammed Aassila
author: Aissa Guesmia
reviewed by: István Ecsedi
MaRDI profile type: Publication
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full work available at URL: s0893-9659(98)00171-2
OpenAlex ID: W1965713660
description: scientific article; zbMATH DE number 1369372