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

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.

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.

Stability for the Mixed Problem Involving the Wave Equation, with Localized Damping, in Unbounded Domains with Finite Measure

2018 ◽  
Vol 56 (4) ◽  
pp. 2802-2834 ◽  
Author(s):  
Marcelo M. Cavalcanti ◽  
Flávio R. Dias Silva ◽  
Valéria N. Domingos Cavalcanti ◽  
André Vicente
Keyword(s):  
Wave Equation ◽  
Mixed Problem ◽  
Unbounded Domains ◽  
Finite Measure ◽  
Localized Damping

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.

GLOBAL EXISTENCE AND ASYMPTOTIC STABILITY FOR THE NONLINEAR AND GENERALIZED DAMPED EXTENSIBLE PLATE EQUATION

2004 ◽  
Vol 06 (05) ◽  
pp. 705-731 ◽  
Author(s):  
M. M. CAVALCANTI ◽  
V. N. DOMINGOS CAVALCANTI ◽  
J. A. SORIANO
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Global Solutions ◽  
Finite Measure ◽  
Decay Rates ◽  
Beam Equation ◽  
Uniform Decay ◽  
Open Set ◽  
Priori Estimates ◽  
The Fixed Point Theorem

The nonlinear and damped extensible plate (or beam) equation is considered [Formula: see text] where Ω is any bounded or unbounded open set of Rn, α>0 and f, g are power like functions. The existence of global solutions is proved by means of the Fixed Point Theorem and continuity arguments. To this end we avoid handling the nonlinearity M(∫Ω|∇u|2dx) in the a priori estimates of energy. Furthermore, uniform decay rates of the energy are also obtained by making use of the perturbed energy method for domains with finite measure.

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 ].

scholarly journals On Existence, Uniform Decay Rates, and Blow-Up for Solutions of a Nonlinear Wave Equation with Dissipative and Source

2012 ◽  
Vol 2012 ◽  
pp. 1-27
Author(s):  
Xiaopan Liu
Keyword(s):  
Wave Equation ◽  
Global Existence ◽  
Nonlinear Wave Equation ◽  
Energy Method ◽  
Blow Up ◽  
Existence Of Solution ◽  
Decay Rates ◽  
Nonlinear Hyperbolic Equation ◽  
Asymptotic Behaviors ◽  
Uniform Decay

This paper studies the blow-up and existence, and asymptotic behaviors of the solution of a nonlinear hyperbolic equation with dissipative and source terms. By using Galerkin procedure and the perturbed energy method, the local and global existence of solution is established. In addition, by the concave method, the blow-up of solutions can be obtained.

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
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