Optimal decay rates for the viscoelastic wave equation

2017 ◽  
Vol 41 (1) ◽  
pp. 192-204 ◽  
Author(s):  
Muhammad I. Mustafa
Keyword(s):  
Wave Equation ◽  
Decay Rates ◽  
Viscoelastic Wave Equation ◽  
Optimal Decay Rates ◽  
Optimal Decay ◽  
Viscoelastic Wave

scholarly journals One-dimensional wave equation with set-valued boundary damping: well-posedness, asymptotic stability, and decay rates

2021 ◽  
Author(s):  
Yacine Chitour ◽  
swann marx ◽  
guilherme mazanti
Keyword(s):  
Wave Equation ◽  
Asymptotic Stability ◽  
Existence And Uniqueness ◽  
Strong Stability ◽  
Decay Rates ◽  
Uniqueness Of Solutions ◽  
Boundary Damping ◽  
Discrete Time Dynamical System ◽  
Optimal Decay Rates ◽  
Optimal Decay

This paper is concerned with the analysis of a 1D wave equation $z_{tt}-z_{xx}=0$ on $[0,1]$ with a Dirichlet condition at $x=0$ and a damping acting at $x=1$ of the form  $(z_t(t,1),-z_x(t,1))\in\Sigma$ for $t\geq 0$, where $\Sigma$ is a given subset of $\mathbb R^2$. The study is performed within an $L^p$ functional framework, $p\in [1, +\infty]$. We determine conditions on $\Sigma$ ensuring existence and uniqueness of solutions of that wave equation,  its strong stability and uniform global asymptotic stability of the solutions. In the latter case, we study the corresponding decay rates  and their optimality. We first establish a correspondence between the solutions of that wave equation and the iterated sequences of a discrete-time dynamical system in terms of which we investigate the above mentioned issues. This enables us to provide a  necessary and sufficient condition on $\Sigma$ ensuring existence and uniqueness of solutions of the wave equation and an efficient strategy for determining optimal decay rates when $\Sigma$ verifies a generalized sector condition.  In case the boundary damping is subject to perturbations, we derive sharp results regarding asymptotic perturbation rejection and input-to-state issues.

scholarly journals A dissipative logarithmic-Laplacian type of plate equation: Asymptotic profile and decay rates

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ruy Coimbra Charão ◽  
Alessandra Piske ◽  
Ryo Ikehata
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Decay Rates ◽  
Plate Equation ◽  
New Model ◽  
Asymptotic Profile ◽  
Optimal Decay Rates ◽  
Optimal Decay ◽  
The Cauchy Problem ◽  
Logarithmic Type

<p style='text-indent:20px;'>We introduce a new model of the logarithmic type of wave like plate equation with a nonlocal logarithmic damping mechanism. We consider the Cauchy problem for this new model in <inline-formula><tex-math id="M1">\begin{document}$ {{\bf R}}^{n} $\end{document}</tex-math></inline-formula>, and study the asymptotic profile and optimal decay rates of solutions as <inline-formula><tex-math id="M2">\begin{document}$ t \to \infty $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M3">\begin{document}$ L^{2} $\end{document}</tex-math></inline-formula>-sense. The operator <inline-formula><tex-math id="M4">\begin{document}$ L $\end{document}</tex-math></inline-formula> considered in this paper was first introduced to dissipate the solutions of the wave equation in the paper studied by Charão-Ikehata [<xref ref-type="bibr" rid="b7">7</xref>]. We will discuss the asymptotic property of the solution as time goes to infinity to our Cauchy problem, and in particular, we classify the property of the solutions into three parts from the viewpoint of regularity of the initial data, that is, diffusion-like, wave-like, and both of them.</p>

scholarly journals A new stability result for a thermoelastic Bresse system with viscoelastic damping

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Soh Edwin Mukiawa ◽  
Cyril Dennis Enyi ◽  
Tijani Abdulaziz Apalara
Keyword(s):  
Stability Result ◽  
Bending Moment ◽  
Decay Rates ◽  
Viscoelastic Damping ◽  
Physical Parameters ◽  
Bresse System ◽  
Solution Energy ◽  
Optimal Decay Rates ◽  
Optimal Decay ◽  
Weaker Conditions

AbstractWe investigate a thermoelastic Bresse system with viscoelastic damping acting on the shear force and heat conduction acting on the bending moment. We show that with weaker conditions on the relaxation function and physical parameters, the solution energy has general and optimal decay rates. Some examples are given to illustrate the findings.

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