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

2007 ◽  
Vol 67 (2) ◽  
pp. 512-544 ◽  
Author(s):  
Daniel Toundykov
Keyword(s):  
Wave Equation ◽  
Boundary Conditions ◽  
Critical Exponent ◽  
Nonlinear Wave Equation ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Decay Rates ◽  
Nonlinear Dissipation ◽  
Optimal Decay Rates ◽  
Optimal Decay

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.

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