scholarly journals Resolvent estimates and the decay of the solution to the wave equation with potential

2001 ◽  
pp. 1-7
Author(s):  
Vladimir Georgiev
Keyword(s):  
Wave Equation ◽  
Resolvent Estimates

scholarly journals Applications of Cutoff Resolvent Estimates to the Wave Equation

2009 ◽  
Vol 16 (4) ◽  
pp. 577-590 ◽  
Author(s):  
Hans Christianson
Keyword(s):  
Wave Equation ◽  
Resolvent Estimates

High Frequency Resolvent Estimates and Energy Decay of Solutions to the Wave Equation

2004 ◽  
Vol 47 (4) ◽  
pp. 504-514 ◽  
Author(s):  
Fernando Cardoso ◽  
Georgi Vodev
Keyword(s):  
Wave Equation ◽  
Real Axis ◽  
Riemannian Manifolds ◽  
High Frequency ◽  
Beltrami Operator ◽  
Energy Decay ◽  
Local Energy ◽  
Resolvent Estimates ◽  
The Real ◽  
Decay Of Solutions

AbstractWe prove an uniform Hölder continuity of the resolvent of the Laplace-Beltrami operator on the real axis for a class of asymptotically Euclidean Riemannian manifolds. As an application we extend a result of Burq on the behaviour of the local energy of solutions to the wave equation.

scholarly journals LOCAL DECAY FOR THE DAMPED WAVE EQUATION IN THE ENERGY SPACE

2016 ◽  
Vol 17 (3) ◽  
pp. 509-540 ◽  
Author(s):  
Julien Royer
Keyword(s):  
Wave Equation ◽  
Long Range ◽  
Short Range ◽  
Absorption Index ◽  
Local Energy ◽  
Damped Wave Equation ◽  
Resolvent Estimates ◽  
Limiting Absorption Principle ◽  
Weighted Energy ◽  
Range Perturbation

We improve a previous result about the local energy decay for the damped wave equation on $\mathbb{R}^{d}$. The problem is governed by a Laplacian associated with a long-range perturbation of the flat metric and a short-range absorption index. Our purpose is to recover the decay ${\mathcal{O}}(t^{-d+\unicode[STIX]{x1D700}})$ in the weighted energy spaces. The proof is based on uniform resolvent estimates, given by an improved version of the dissipative Mourre theory. In particular, we have to prove the limiting absorption principle for the powers of the resolvent with inserted weights.

scholarly journals OBSERVABILITY OF BAOUENDI–GRUSHIN-TYPE EQUATIONS THROUGH RESOLVENT ESTIMATES

2021 ◽  
pp. 1-39
Author(s):  
Cyril Letrouit ◽  
Chenmin Sun
Keyword(s):  
Wave Equation ◽  
Decay Rate ◽  
Evolution Equations ◽  
Type Equation ◽  
Small Time ◽  
Type Operator ◽  
Resolvent Estimate ◽  
Damped Wave Equation ◽  
Resolvent Estimates ◽  
Speed Of Propagation

Abstract In this article, we study the observability (or equivalently, the controllability) of some subelliptic evolution equations depending on their step. This sheds light on the speed of propagation of these equations, notably in the ‘degenerated directions’ of the subelliptic structure. First, for any $\gamma \geq 1$ , we establish a resolvent estimate for the Baouendi–Grushin-type operator $\Delta _{\gamma }=\partial _x^2+\left \lvert x\right \rvert ^{2\gamma }\partial _y^2$ , which has step $\gamma +1$ . We then derive consequences for the observability of the Schrödinger-type equation $i\partial _tu-\left (-\Delta _{\gamma }\right )^{s}u=0$ , where $s\in \mathbb N$ . We identify three different cases: depending on the value of the ratio $(\gamma +1)/s$ , observability may hold in arbitrarily small time or only for sufficiently large times or may even fail for any time. As a corollary of our resolvent estimate, we also obtain observability for heat-type equations $\partial _tu+\left (-\Delta _{\gamma }\right )^su=0$ and establish a decay rate for the damped wave equation associated with $\Delta _{\gamma }$ .

scholarly journals Homogenisation with error estimates of attractors for damped semi-linear anisotropic wave equations

2019 ◽  
Vol 9 (1) ◽  
pp. 745-787 ◽  
Author(s):  
Shane Cooper ◽  
Anton Savostianov
Keyword(s):  
Wave Equation ◽  
Error Estimates ◽  
Wave Equations ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Resolvent Estimates ◽  
First Order ◽  
Sharp Estimates ◽  
Natural Energy ◽  
First Order Correction

Abstract Homogenisation of global 𝓐ε and exponential 𝓜ε attractors for the damped semi-linear anisotropic wave equation $\begin{array}{} \displaystyle \partial_t ^2u^\varepsilon + y \partial_t u^\varepsilon-\operatorname{div} \left(a\left( \tfrac{x}{\varepsilon} \right)\nabla u^\varepsilon \right)+f(u^\varepsilon)=g, \end{array}$ on a bounded domain Ω ⊂ ℝ3, is performed. Order-sharp estimates between trajectories uε(t) and their homogenised trajectories u0(t) are established. These estimates are given in terms of the operator-norm difference between resolvents of the elliptic operator $\begin{array}{} \displaystyle \operatorname{div}\left(a\left( \tfrac{x}{\varepsilon} \right)\nabla \right) \end{array}$ and its homogenised limit div (ah∇). Consequently, norm-resolvent estimates on the Hausdorff distance between the anisotropic attractors and their homogenised counter-parts 𝓐0 and 𝓜0 are established. These results imply error estimates of the form distX(𝓐ε, 𝓐0) ≤ Cεϰ and $\begin{array}{} \displaystyle \operatorname{dist}^s_X(\mathcal M^\varepsilon, \mathcal M^0) \le C \varepsilon^\varkappa \end{array}$ in the spaces X = L2(Ω) × H–1(Ω) and X = (Cβ(Ω))2. In the natural energy space 𝓔 := $\begin{array}{} \displaystyle H^1_0 \end{array}$(Ω) × L2(Ω), error estimates dist𝓔(𝓐ε, Tε 𝓐0) ≤ $\begin{array}{} \displaystyle C \sqrt{\varepsilon}^\varkappa \end{array}$ and $\begin{array}{} \displaystyle \operatorname{dist}^s_\mathcal{E}(\mathcal M^\varepsilon, \text{T}_\varepsilon \mathcal M^0) \le C \sqrt{\varepsilon}^\varkappa \end{array}$ are established where Tε is first-order correction for the homogenised attractors suggested by asymptotic expansions. Our results are applied to Dirchlet, Neumann and periodic boundary conditions.

Extended F-Expansion Method for Solving the modified Korteweg-de Vries (mKdV) Equation

2020 ◽  
Vol 11 (1) ◽  
pp. 93-100
Author(s):  
Vina Apriliani ◽  
Ikhsan Maulidi ◽  
Budi Azhari
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Internal Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Mkdv Equation ◽  
Innovative Methods ◽  
De Vries

One of the phenomenon in marine science that is often encountered is the phenomenon of water waves. Waves that occur below the surface of seawater are called internal waves. One of the mathematical models that can represent solitary internal waves is the modified Korteweg-de Vries (mKdV) equation. Many methods can be used to construct the solution of the mKdV wave equation, one of which is the extended F-expansion method. The purpose of this study is to determine the solution of the mKdV wave equation using the extended F-expansion method. The result of solving the mKdV wave equation is the exact solutions. The exact solutions of the mKdV wave equation are expressed in the Jacobi elliptic functions, trigonometric functions, and hyperbolic functions. From this research, it is expected to be able to add insight and knowledge about the implementation of the innovative methods for solving wave equations. 

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