Related papers: Quantitative Mode Stability for the Wave Equation …
We discuss the dynamics of the quantum fluctuation around the nonlinear massive wave solution in the Higgs potential. In particular, we analyze the stability and instability of the mode function. Using the stability condition for Hill's…
In this paper we consider a stabilization problem for the abstract-wave equation with delay. We prove an exponential stability result for appropriate damping coefficient. The proof of the main result is based on a frequency-domain approach.
It is shown that the properties of the modulational instability of partially coherent waves propagating in a nonlinear Kerr medium depend crucially on the profile of the incoherent field spectrum. Under certain conditions, the incoherence…
We prove the existence of instabilities for the geometric linear wave equation on extremal Kerr spacetime backgrounds, which describe stationary black holes rotating at their maximally allowed angular velocity. These instabilities can be…
We investigate a two-dimensional transmission model consisting of a wave equation and a Kirchhoff plate equation with dynamical boundary controls under geometric conditions. The two equations are coupled through transmission conditions…
We study axisymmetric solutions to the wave equation on extremal Kerr backgrounds and obtain integrated local energy decay (or Morawetz estimates) through an analysis \textit{exclusively in physical-space}. Boundedness of the energy and…
This paper is concerned with the decay estimate of solutions to the semilinear wave equation subject to two localized dampings in a bounded domain. The first one is of the nonlinear Kelvin-Voigt type and is distributed around a neighborhood…
In this article we study the pointwise decay properties of solutions to the wave equation on a class of stationary asymptotically flat backgrounds in three space dimensions. Under the assumption that uniform energy bounds and a weak form of…
We study the dispersive properties for the wave equation in the Kerr space-time with small angular momentum. The main result of this paper is to establish Strichartz estimates for solutions of the aforementioned equation. As an application,…
In this paper, we investigate the stabilization of a locally coupled wave equations with local viscoelastic damping of past history type acting only in one equation via non smooth coefficients. First, using a general criteria of…
Continuing the investigations started in the recent work [Krieger-Xiang, 2022] on semi-global controllability and stabilization of the $(1+1)$-dimensional wave maps equation with spatial domain $\mathbb{S}^1$ and target $\mathbb{S}^k$,…
We prove local and global energy decay for the wave equation in a wave guide with damping at infinity. More precisely, the absorption index is assumed to converge slowly to a positive constant, and we obtain the diffusive phenomenon typical…
We study a damped semi-linear wave equation in a bounded domain with smooth boundary. It is proved that any sufficiently smooth solution can be stabilised locally by a finite-dimensional feedback control supported by a given open subset…
Using the density matrix formalism, we prove an existence theorem of the periodic steady-state for an arbitrary periodically-driven system. This state has the same period as the modulated external influence, and it is realized as an…
We investigate the stability of the wave equation with spatial dependent coefficients on a bounded multidimensional domain. The system is stabilized via a scattering passive feedback law. We formulate the wave equation in a port-Hamiltonian…
Solutions for a class of wave equations with effective potentials are obtained by a method of a Laplace-transform. Quasinormal modes appear naturally in the solutions only in a spatially truncated form; their coefficients are uniquely…
This is our main paper in a series in which we prove the full, unconditional, nonlinear stability of the Kerr family $Kerr(a, m)$ for small angular momentum, i.e. $|a|/m\ll 1$, in the context of asymptotically flat solutions of the Einstein…
The first article in a two-part series (the second article being [arXiv:2205.13197]) assumes a weak local energy decay estimate holds and proves that solutions to the linear wave equation with variable coefficients in $\mathbb R^{1+3}$,…
For fractional wave equations with low H\"older regularity damping, we establish quantitative energy decay rates for their solutions when the geometric control condition holds. The energy decay rates depend explicitly on the H\"older…
We consider stability of periodic travelling waves in the generalized reduced Ostrovsky equation with respect to co-periodic perturbations. Compared to the recent literature, we give a simple argument that proves spectral stability of all…