Uniform Decay Rates for Viscoelastic Dissipative Systems

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.

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

Advanced Nonlinear Studies ◽

10.1515/ans-2021-2150 ◽

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.

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Uniform decay rates of a coupled suspension bridges with temperature

Arabian Journal of Mathematics ◽

10.1007/s40065-021-00342-0 ◽

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.

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Wave Propagation on Microstate Geometries

Annales Henri Poincaré ◽

10.1007/s00023-019-00874-4 ◽

2019 ◽

Vol 21 (3) ◽

pp. 705-760 ◽

Cited By ~ 5

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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