Exponential stability for the wave model with localized memory in a past history framework
type: Item
instance of: scholarly article
title: Exponential stability for the wave model with localized memory in a past history framework
zbMATH Open document ID: 1404.35034
author: A. Y. de Souza Franco
author: M. M. Cavalcanti
author: V. N. Domingos Cavalcanti
author: M. A. Jorge Silva
published in: Journal of Differential Equations
publication date: 2018-03-22T00:00:00Z
review text: The article investigates the well-posedness as well as the exponential decay of solutions to a mixed problem of a locally distributed viscoelastic damped wave equation in the variable u(x,t), with past history. Introducing a new variable \(\eta^t(x,s) = u(x,t)-u(x,t-s)\) the model is transformed into a mixed problem for a system in variables \(u\) and \(\eta\), \[ \begin{aligned} &{\rho}(x) u_{tt}-\operatorname{div}[k(x)\nabla u]-\int_0^{\infty}g(s)\operatorname{div}a(x)\nabla{\eta}^t(.,s)]\,\mathrm ds+b(x)u_t=0,\quad \text{in }\Omega \times (0,\infty)\\ &\eta_t=-\eta_s+u_t \qquad\text{in }\Omega \times(0,\infty)\times (0,\infty),\\ &u=0 \quad \text{ on } \partial\Omega \times (0,\infty), \;\eta=0 \quad \text{ on } \partial\Omega \times (0,\infty),\\ &u(x,0)=u_0(x), \; u_t(x,0)=u_1(x), \\ &\eta^0(x,s)=\eta_0(x,s), \qquad x\in\Omega, \;s\in[0,\infty), \\ &\eta^t(x,0) =0, \qquad x\in\Omega, \;t\in[0,\infty) \end{aligned} \] To this system the energy functional \({E_{u,{\eta}}}(t)\) is associated. In Section 2, after a quite extensive review of the literature on well-posedness and decay properties of solutions to the wave equation, under some assumptions, a well-posedness result for the above system is proved. From it, a well posedness result for the initial mixed problem is deduced. In the third section, under additional assumptions, an exponential stability result is proved and one investigates particular cases when system is exponentially stable in the absence of the frictional damping term. The forth section is devoted to a result related to the local uniform decay associated with the semi-linear model, which is the above model, where a nonlinear source \(f(u)\) has been introduced. The fifth section is the Appendix and contains some additional definitions and results that might be used in the proofs of exponential stability.
reviewed by: Claudia Simionescu-Badea
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35A27
Mathematics Subject Classification ID: 74Dxx
Mathematics Subject Classification ID: 35L20
zbMATH DE Number: 6852245
zbMATH Keywords: exponential decay
zbMATH Keywords: locally distributed viscoelastic damping
MaRDI profile type: Publication
full work available at URL: j.jde.2018.01.044
OpenAlex ID: W2789534274
cites work: Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems
cites work: A general method for proving sharp energy decay rates for memory-dissipative evolution equations
cites work: Decay estimates for second order evolution equations with memory
cites work: ON EXPONENTIAL ASYMPTOTIC STABILITY IN LINEAR VISCOELASTICITY
cites work: Sharp Sufficient Conditions for the Observation, Control, and Stabilization of Waves from the Boundary
cites work: Decay of solutions of the elastic wave equation with a localized dissipation
cites work: Uniform stabilization of the wave equation on compact surfaces and locally distributed damping
cites work: Asymptotic stability of the wave equation on compact surfaces and locally distributed damping-A sharp result
cites work: Asymptotic stability of the wave equation on compact manifolds and locally distributed damping: A sharp result
cites work: Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects
cites work: Attractors for wave equations with degenerate memory
cites work: Frictional versus Viscoelastic Damping in a Semilinear Wave Equation
cites work: A well posedness result for nonlinear viscoelastic equations with memory
cites work: Global attractors for nonlinear viscoelastic equations with memory
cites work: Nonclassical diffusion with memory
cites work: Asymptotic stability in viscoelasticity
cites work: Q3322454
cites work: Exponential attractors for abstract equations with memory and applications to viscoelasticity
cites work: Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions
cites work: Stabilization and control for the subcritical semilinear wave equation
cites work: A new approach to equations with memory
cites work: Microlocal defect measures
cites work: Asymptotic behavior of a semilinear problem in heat conduction with memory
cites work: Global attractors for a semilinear hyperbolic equation in viscoelasticity
cites work: Q4407520
cites work: A general decay result for a viscoelastic equation in the presence of past and finite history memories
cites work: Q3150052
cites work: Stabilization for the semilinear wave equation with geometric control condition
cites work: Riemannian geometry and geometric analysis
cites work: Dispersive estimates for principally normal pseudodifferential operators
cites work: Note on intrinsic decay rates for abstract wave equations with memory
cites work: Q4879447
cites work: Exponential decay of energy of vibrating strings with local viscoelasticity
cites work: A new method to obtain decay rate estimates for dissipative systems with localized damping
cites work: Escape Function Conditions for the Observation, Control, and Stabilization of the Wave Equation
cites work: Asymptotic behaviour of the energy in partially viscoelastic materials
cites work: Decay and Global Existence for Nonlinear Wave Equations with Localized Dissipations in General Exterior Domains
cites work: Energy decay for the wave equation with boundary and localized dissipations in exterior domains
cites work: Stability and exponential stability in linear viscoelasticity
cites work: Semigroups of linear operators and applications to partial differential equations
cites work: Asymptotic behavior of a class of abstract semilinear integrodifferential equations and applications
cites work: Solutions of the wave equation with localized energy
cites work: Decay of solutions to nondissipative hyperbolic systems on compact manifolds
cites work: Optimal decay rates for solutions of a nonlinear wave equation with localized nonlinear dissipation of unrestricted growth and critical exponent source terms under mixed boundary conditions
cites work: Carleman estimates with no lower-order terms for general Riemann wave equations. Global uniqueness and observability in one shot
cites work: Hadamard well-posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping
cites work: Exponential Decay for The Semilinear Wave Equation with Locally Distributed Damping
DOI: 10.1016/J.JDE.2018.01.044
description: scientific article; zbMATH DE number 6852245