Uniform decay rates of coupled anisotropic elastodynamic/Maxwell equations with nonlinear damping

2011 ◽  
pp. 205-238
Author(s):  
Cleverson Roberto da Luz ◽  
Gustavo Alberto Perla Menzala
Keyword(s):  
Maxwell Equations ◽  
Nonlinear Damping ◽  
Decay Rates ◽  
Uniform Decay

Well-Posedness and Uniform Decay Rates for a Nonlinear Damped Schrödinger-Type Equation

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Marcelo M. Cavalcanti ◽  
Valéria N. Domingos Cavalcanti
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Differential Operators ◽  
Type Equation ◽  
Nonlinear Damping ◽  
Decay Rates ◽  
Dirichlet Boundary Conditions ◽  
Uniform Decay ◽  
Pseudo Differential Operators ◽  
The One

Abstract In this paper we study the existence as well as uniform decay rates of the energy associated with the nonlinear damped Schrödinger equation, i ⁢ u t + Δ ⁢ u + | u | α ⁢ u - g ⁢ ( u t ) = 0   in  ⁢ Ω × ( 0 , ∞ ) , iu_{t}+\Delta u+|u|^{\alpha}u-g(u_{t})=0\quad\text{in }\Omega\times(0,\infty), subject to Dirichlet boundary conditions, where Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} , n ≤ 3 {n\leq 3} , is a bounded domain with smooth boundary ∂ ⁡ Ω = Γ {\partial\Omega=\Gamma} and α = 2 , 3 {\alpha=2,3} . Our goal is to consider a different approach than the one used in [B. Dehman, P. Gérard and G. Lebeau, Stabilization and control for the nonlinear Schrödinger equation on a compact surface, Math. Z. 254 2006, 4, 729–749], so instead than using the properties of pseudo-differential operators introduced by cited authors, we consider a nonlinear damping, so that we remark that no growth assumptions on g ⁢ ( z ) {g(z)} are made near the origin.

Uniform stabilization for the semilinear wave equation in an inhomogeneous medium with locally distributed nonlinear damping and dynamic Cauchy–Ventcel type boundary conditions

2019 ◽  
pp. 1950072
Author(s):  
A. F. Almeida ◽  
M. M. Cavalcanti ◽  
V. H. Gonzalez Martinez ◽  
J. P. Zanchetta
Keyword(s):  
Boundary Conditions ◽  
Inhomogeneous Medium ◽  
Linear Problem ◽  
Nonlinear Damping ◽  
Decay Rates ◽  
Geometric Control ◽  
Dynamic Boundary Conditions ◽  
Uniform Stabilization ◽  
Uniform Decay ◽  
Type Boundary

In this paper, we consider the Cauchy–Ventcel problem in an inhomogeneous medium with dynamic boundary conditions subject to a nonlinear damping distributed around a neighborhood [Formula: see text] of the boundary according to the Geometric Control Condition. Uniform decay rates of the associated energy are established and, in addition, the exact internal controllability for the linear problem is also proved. For this purpose, refined microlocal analysis arguments are considered by exploiting ideas due to Burq and Gérard [Contrôle Optimal des équations aux dérivées partielles. (2001); http://www.math.u-psud.fr/burq/articles/coursX.pdf ].

Uniform decay rates for the energy of Timoshenko system with the arbitrary speeds of propagation and localized nonlinear damping

2013 ◽  
Vol 65 (6) ◽  
pp. 1189-1206 ◽  
Author(s):  
M. M. Cavalcanti ◽  
V. N. Domingos Cavalcanti ◽  
F. A. Falcão Nascimento ◽  
I. Lasiecka ◽  
J. H. Rodrigues
Keyword(s):  
Nonlinear Damping ◽  
Decay Rates ◽  
Timoshenko System ◽  
Uniform Decay

Uniform Decay Rates for the Wave Equation with Nonlinear Damping Locally Distributed in Unbounded Domains with Finite Measure

2014 ◽  
Vol 52 (1) ◽  
pp. 545-580 ◽  
Author(s):  
Marcelo M. Cavalcanti ◽  
Flávio R. Dias Silva ◽  
Valéria N. Domingos Cavalcanti
Keyword(s):  
Wave Equation ◽  
Unbounded Domains ◽  
Finite Measure ◽  
Nonlinear Damping ◽  
Decay Rates ◽  
Uniform Decay

scholarly journals Uniform decay rates for a suspension bridge with locally distributed nonlinear damping

2020 ◽  
Vol 357 (4) ◽  
pp. 2388-2419 ◽  
Author(s):  
André D. Domingos Cavalcanti ◽  
Marcelo M. Cavalcanti ◽  
Wellington J. Corrêa ◽  
Zayd Hajjej ◽  
Mauricio Sepúlveda Cortés. ◽  
...  
Keyword(s):  
Suspension Bridge ◽  
Nonlinear Damping ◽  
Decay Rates ◽  
Uniform Decay

Stabilization of the wave equation with a nonlinear delay term in the boundary conditions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Wassila Ghecham ◽  
Salah-Eddine Rebiai ◽  
Fatima Zohra Sidiali
Keyword(s):  
Wave Equation ◽  
Original Problem ◽  
Existence Of Solutions ◽  
Semigroup Theory ◽  
Decay Rates ◽  
Uniform Decay ◽  
Delay Term ◽  
Nonlinear Semigroup Theory ◽  
Nonlinear Boundary Feedback ◽  
Nonlinear Delay

Abstract A wave equation in a bounded and smooth domain of ℝ n {\mathbb{R}^{n}} with a delay term in the nonlinear boundary feedback is considered. Under suitable assumptions, global existence and uniform decay rates for the solutions are established. The proof of existence of solutions relies on a construction of suitable approximating problems for which the existence of the unique solution will be established using nonlinear semigroup theory and then passage to the limit gives the existence of solutions to the original problem. The uniform decay rates for the solutions are obtained by proving certain integral inequalities for the energy function and by establishing a comparison theorem which relates the asymptotic behavior of the energy and of the solutions to an appropriate dissipative ordinary differential equation.

scholarly journals Uniform decay rates of a coupled suspension bridges with temperature

2021 ◽  
Author(s):  
Mounir Afilal ◽  
Mohamed Alahyane ◽  
Abdelaziz Soufyane
Keyword(s):  
Energy Method ◽  
Lyapunov Functional ◽  
Decay Rates ◽  
Suspension Bridges ◽  
Uniform Decay ◽  
Decay Properties

AbstractIn this paper, we investigate the decay properties of the thermoelastic suspension bridges model. We prove that the energy is decaying exponentially. To our knowledge, our result is new and our method of proof is based on the energy method to build the appropriate Lyapunov functional.

Uniform Decay Rates for Viscoelastic Dissipative Systems

2014 ◽  
Vol 22 (1) ◽  
pp. 101-116 ◽  
Author(s):  
Muhammad I. Mustafa
Keyword(s):  
Decay Rates ◽  
Dissipative Systems ◽  
Uniform Decay

scholarly journals Wave Propagation on Microstate Geometries

2019 ◽  
Vol 21 (3) ◽  
pp. 705-760 ◽  
Author(s):  
Joe Keir
Keyword(s):  
Wave Equation ◽  
Decay Rates ◽  
Mathematical Treatment ◽  
Linear Wave ◽  
Local Energy ◽  
Linear Wave Equation ◽  
Uniform Decay ◽  
Uniformly Bounded ◽  
Kaluza Klein ◽  
Made In

AbstractSupersymmetric microstate geometries were recently conjectured (Eperon et al. in JHEP 10:031, 2016. 10.1007/JHEP10(2016)031) to be nonlinearly unstable due to numerical and heuristic evidence, based on the existence of very slowly decaying solutions to the linear wave equation on these backgrounds. In this paper, we give a thorough mathematical treatment of the linear wave equation on both two- and three-charge supersymmetric microstate geometries, finding a number of surprising results. In both cases, we prove that solutions to the wave equation have uniformly bounded local energy, despite the fact that three-charge microstates possess an ergoregion; these geometries therefore avoid Friedman’s “ergosphere instability” (Friedman in Commun Math Phys 63(3):243–255, 1978). In fact, in the three-charge case we are able to construct solutions to the wave equation with local energy that neither grows nor decays, although these data must have non-trivial dependence on the Kaluza–Klein coordinate. In the two-charge case, we construct quasimodes and use these to bound the uniform decay rate, showing that the only possible uniform decay statements on these backgrounds have very slow decay rates. We find that these decay rates are sublogarithmic, verifying the numerical results of Eperon et al. (2016). The same construction can be made in the three-charge case, and in both cases the data for the quasimodes can be chosen to have trivial dependence on the Kaluza–Klein coordinates.

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