Exponential decay for the semilinear wave equation with localized frictional and Kelvin–Voigt dissipating mechanisms

2021 ◽  
pp. 1-21
Author(s):  
Marcelo M. Cavalcanti ◽  
Victor H. Gonzalez Martinez
Keyword(s):  
Wave Equation ◽  
Phase Space ◽  
Initial Data ◽  
Inhomogeneous Medium ◽  
Finite Energy ◽  
Geometric Control ◽  
Geometric Condition ◽  
Frictional Damping ◽  
Viscoelastic Wave ◽  
Bounded Sets

In the present paper, we are concerned with the semilinear viscoelastic wave equation in an inhomogeneous medium Ω subject to two localized dampings. The first one is of the type viscoelastic and is distributed around a neighborhood ω of the boundary according to the Geometric Control Condition. The second one is a frictional damping and we consider it hurting the geometric condition of control. We show that the energy of the wave equation goes uniformly and exponentially to zero for all initial data of finite energy taken in bounded sets of finite energy phase-space.

scholarly journals Blow-Up and Global Existence Analysis for the Viscoelastic Wave Equation with a Frictional and a Kelvin-Voigt Damping

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Fosheng Wang ◽  
Chengqiang Wang
Keyword(s):  
Wave Equation ◽  
Boundary Value Problem ◽  
Blow Up ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Viscoelastic Wave Equation ◽  
Frictional Damping ◽  
Viscoelastic Wave ◽  
Initial Boundary Value

We are concerned in this paper with the initial boundary value problem for a quasilinear viscoelastic wave equation which is subject to a nonlinear action, to a nonlinear frictional damping, and to a Kelvin-Voigt damping, simultaneously. By utilizing a carefully chosen Lyapunov functional, we establish first by the celebrated convexity argument a finite time blow-up criterion for the initial boundary value problem in question; we prove second by an a priori estimate argument that some solutions to the problem exists globally if the nonlinearity is “weaker,” in a certain sense, than the frictional damping, and if the viscoelastic damping is sufficiently strong.

General decay results for a viscoelastic wave equation with a variable exponent nonlinearity

2021 ◽  
pp. 1-24
Author(s):  
Jamilu Hashim Hassan ◽  
Salim A. Messaoudi
Keyword(s):  
Wave Equation ◽  
Decay Rate ◽  
Relaxation Function ◽  
Variable Exponent ◽  
General Decay ◽  
Viscoelastic Wave Equation ◽  
Frictional Damping ◽  
Viscoelastic Wave

In this paper we consider a viscoelastic wave equation with a very general relaxation function and nonlinear frictional damping of variable-exponent type. We give explicit and general decay results for the energy of the system depending on the decay rate of the relaxation function and the nature of the variable-exponent nonlinearity. Our results extend the existing results in the literature to the case of nonlinear frictional damping of variable-exponent type.

scholarly journals Exponential decay for the damped wave equation in unbounded domains

2016 ◽  
Vol 18 (06) ◽  
pp. 1650012 ◽  
Author(s):  
Nicolas Burq ◽  
Romain Joly
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Exponential Decay ◽  
Unbounded Domain ◽  
Unbounded Domains ◽  
Geometric Control ◽  
Damped Wave Equation ◽  
Logarithmic Decay ◽  
And Control

We study the decay of the semigroup generated by the damped wave equation in an unbounded domain. We first prove under the natural geometric control condition the exponential decay of the semigroup. Then we prove under a weaker condition the logarithmic decay of the solutions (assuming that the initial data are smoother). As corollaries, we obtain several extensions of previous results of stabilization and control.

scholarly journals Existence and general decay of Balakrishnan-Taylor viscoelastic equation with nonlinear frictional damping and logarithmic source term

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Mohammad Al-Gharabli ◽  
Mohamed Balegh ◽  
Baowei Feng ◽  
Zayd Hajjej ◽  
Salim A. Messaoudi
Keyword(s):  
Wave Equation ◽  
Convex Functions ◽  
General Type ◽  
Source Term ◽  
General Decay ◽  
General Decay Rate ◽  
Relaxation Functions ◽  
Frictional Damping ◽  
Viscoelastic Wave ◽  
Viscoelastic Equation

<p style='text-indent:20px;'>In this paper, we consider a Balakrishnan-Taylor viscoelastic wave equation with nonlinear frictional damping and logarithmic source term. By assuming a more general type of relaxation functions, we establish explicit and general decay rate results, using the multiplier method and some properties of the convex functions. This result is new and generalizes earlier results in the literature.</p>

scholarly journals Hamiltonian charges in the asymptotically de Sitter spacetimes

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Maciej Kolanowski ◽  
Jerzy Lewandowski
Keyword(s):  
Phase Space ◽  
Initial Data ◽  
Exact Solutions ◽  
Gravitational Waves ◽  
Physical Meaning ◽  
De Sitter ◽  
Killing Vectors ◽  
Asymptotically Flat ◽  
Angular Momenta ◽  
The One

Abstract We generalize a notion of ‘conserved’ charges given by Wald and Zoupas to the asymptotically de Sitter spacetimes. Surprisingly, our construction is less ambiguous than the one encountered in the asymptotically flat context. An expansion around exact solutions possessing Killing vectors provides their physical meaning. In particular, we discuss a question of how to define energy and angular momenta of gravitational waves propagating on Kottler and Carter backgrounds. We show that obtained expressions have a correct limit as Λ → 0. We also comment on the relation between this approach and the one based on the canonical phase space of initial data at ℐ+.

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