This paper is devoted to the derivation of $L^2$ - $L^2$ decay estimates for the solution of the homogeneous linear damped wave equation on the Heisenberg group $\mathbf{H}_n$, for its time derivative and for its horizontal gradient. Moreover, we consider the improvement of these estimates when further $L^1(\mathbf{H}_n)$ regularity is required for the Cauchy data. Our approach will rely strongly on the group Fourier transform of $\mathbf{H}_n$ and on the properties of the Hermite functions that form a maximal orthonormal system for $L^2(\mathbb{R}^n)$ of eigenfunctions of the harmonic oscillator.
In this paper, we consider the Cauchy problem for the semilinear damped wave equation on the Heisenberg group with power nonlinearity. We prove that the critical exponent is the Fujita exponent $p_{\mathrm{Fuj}}(\mathscr{Q}) = 1+2 / \mathscr{Q}$, where $\mathscr{Q}$ is the homogeneous dimension of the Heisenberg group. On the one hand, we will prove the global existence of small data solutions for $p >p_{\mathrm{Fuj}}(\mathscr{Q})$ in an exponential weighted energy space. On the other hand, a blow-up result for $1 < p \leq p_{\mathrm{Fuj}}(\mathscr{Q})$ under certain integral sign assumptions …
A condition which guaranties the exponential decay of the solutions of the initial-boundary value problem for the damped wave equation is proved. A method for the effective computability of the coefficient of exponential decay is also presented.
We study the decay of the semigroup generated by the damped wave equation in an unbounded domain. We first prove under the natural geometric control condition the exponential decay of the semigroup. Then we prove under a weaker condition the logarithmic decay of the solutions (assuming that the initial data are smoother). As corollaries, we obtain several extensions of previous results of stabilisation and control.
We study in this article decay rates for Kelvin-Voigt damped wave equations under a geometric control condition. We prove that when the damping coefficient is sufficiently smooth ($C^1$ vanishing nicely) we show that exponential decay follows from geometric control conditions (see~\cite{BuCh, Te12} for similar results under stronger assumptions on the damping function).
We derive a uniform exponential decay of the total energy for the nonlinear Klein-Gordon equation with a damping around spatial infinity in the whole space or in the exterior of a star shaped obstacle.
We study the decay of the global energy for the damped Klein-Gordon equation on non-compact manifolds with finitely many cylindrical and subconic ends up to bounded perturbation. We prove that under the Geometric Control Condition, the decay is exponential, and that under the weaker Network Control Condition, the decay is logarithmic, by developing the global Carleman estimate with multiple weights.
In this paper we consider a compact Riemannian manifold (M, g) of class C 1 $\cap$ W 2,$\infty$ and the damped wave or Schrödinger equations on M , under the action of a damping function a = a(x). We establish the following fact: if the measure of the set {x $\in$ M ; a(x) = 0} is strictly positive, then the decay in time of the associated energy is at least logarithmic.
In this note, we study the asymptotic behavior, as $t$ tends to infinity, of the solution $u$ to the evolutionary damped $p$-Laplace equation \begin{equation*} u_{tt}+a\, u_t =Δ_p u \end{equation*} with Dirichlet boundary values. Let $u^*$ denote the stationary solution with same boundary values, then the $W^{1,p}$-norm of $u(t) - u^{*}$ decays for large $t$ like $t^{-\frac{1}{(p-1)p}}$, in the degenerate case $ p > 2$.
The exponential decay rate of $L^2-$norm related to the Korteweg-de Vries equation with localized damping posed on whole real line will be established. In addition, by using classical arguments we determine the $H^1-$norm of the solution associated to Korteweg-de Vries equation with damping in whole domain, can not have a decay property for an arbitrary initial data.
This paper is concerned with decay estimate of solutions to the semilinear wave equation with strong damping in a bounded domain. Introducing an appropriate Lyaponuv function, we prove that when the damping is linear, we can find initial data, for which the solution decays exponentially. This result improves an early one in an article of Gazzola and Squassina.
In this paper we study the Cauchy problem for the semilinear damped wave equation for the sub-Laplacian on the Heisenberg group. In the case of the positive mass, we show the global in time well-posedness for small data for power like nonlinearities. We also obtain similar well-posedness results for the wave equations for Rockland operators on general graded Lie groups. In particular, this includes higher order operators on $\mathbb R^n$ and on the Heisenberg group, such as powers of the Laplacian or of the sub-Laplacian. In addition, we establish a new family of Gagliardo-Nirenberg inequaliti…
We study the Cauchy problem for the nonlinear damped wave equation and establish the large data local well-posedness and small data global well-posedness with slowly decaying initial data. We also prove that the asymptotic profile of the global solution is given by a solution of the corresponding parabolic problem, which shows that the solution of the damped wave equation has the diffusion phenomena. Moreover, we show blow-up of solution and give the estimate of the lifespan for a subcritical nonlinearity. In particular, we determine the critical exponent for any space dimension.
We consider damped wave (resp. Schr{ö}dinger and plate) equations driven by a hypoelliptic "sum of squares" operator L on a compact manifold and a damping function b(x). We assume the Chow-Rashevski-H{ö}rmander condition at rank k (at most k Lie brackets needed to span the tangent space) together with analyticity of M and the coefficients of L. We prove decay of the energy at rate $log(t)^{-1/k}$ (resp. $log(t)^{-2/k}$ ) for data in the domain of the generator of the associated group. We show that this decay is optimal on a family of Grushin-type operators. This result follows from a perturbat…
We show in this article how perturbative approaches~from our work with Hitrik (see also the work by Anantharaman-Macia) and the {\em black box} strategy from~ our work with Zworski allow to obtain decay rates for Kelvin-Voigt damped wave equations from quite standard resolvent estimates : Carleman estimates or geometric control estimates for Helmoltz equationCarleman or other resolvent estimates for the Helmoltz equation. Though in this context of Kelvin Voigt damping, such approach is unlikely to allow for the optimal results when additional geometric assumptions are considered (see \cite{BuC…
We address the decay rates of the energy for the damped wave equation when the damping coefficient $b$ does not satisfy the Geometric Control Condition (GCC). First, we give a link with the controllability of the associated Schrödinger equation. We prove in an abstract setting that the observability of the Schrödinger group implies that the semigroup associated to the damped wave equation decays at rate $1/\sqrt{t}$ (which is a stronger rate than the general logarithmic one predicted by the Lebeau Theorem). Second, we focus on the 2-dimensional torus. We prove that the best decay one can expec…
We consider the semilinear damped wave equation $\partial_{tt}^2 u(x,t)+γ(x)\partial_t u(x,t)=Δu(x,t)-αu(x,t)-f(x,u(x,t))$. In this article, we obtain the first results concerning the stabilization of this semilinear equation in cases where $γ$ does not satisfy the geometric control condition. When some of the geodesic rays are trapped, the stabilization of the linear semigroup is semi-uniform in the sense that $\|e^{At}A^{-1}\|\leq h(t)$ for some function $h$ with $h(t)\rightarrow 0$ when $t\rightarrow +\infty$. We provide general tools to deal with the semilinear stabilization problem in the…
We consider the fractional Klein-Gordon equation in one spatial dimension, subjected to a damping coefficient, which is non-trivial and periodic, or more generally strictly positive on a periodic set. We show that the energy of the solution decays at the polynomial rate $O(t^{-\frac{s}{4-2s}})$ for $0< s<2 $ and at some exponential rate when $s\geq 2$. Our approach is based on the asymptotic theory of $C_0$ semigroups in which one can relate the decay rate of the energy in terms of the resolvent growth of the semigroup generator. The main technical result is a new observability estimate for th…
We cope with a free boundary fluid-structure interaction model. In the model, the viscous incompressible fluid interacts with elastic body via the common boundary. The motion of the fluid is governed by Navier-Stokes equations while the displacement of elastic structure is described by variable coefficient wave equations. The dissipation is placed on the common boundary between fluid and elastic body. Given small initial data, the global existence of the solutions of this system is proved and the exponential decay of solutions are obtained.
In this paper we study a semilinear wave equation with nonlinear, time-dependent damping in one space dimension. For this problem, we prove a well-posedness result in $W^{1,\infty}$ in the space-time domain $(0,1)\times [0,+\infty)$. Then we address the problem of the time-asymptotic stability of the zero solution and show that, under appropriate conditions, the solution decays to zero at an exponential rate in the space $W^{1,\infty}$. The proofs are based on the analysis of the corresponding semilinear system for the first order derivatives, for which we show a contractive property of the in…
Under appropriate assumptions the energy of wave equations with damping and variable coefficients $c(x)u_{tt}-\hbox{div}(b(x)\nabla u)+a(x)u_t =h(x)$ has been shown to decay. Determining the rate of decay for the higher order energies involving the $k$th order spatial and time derivatives has been an open problem with the exception of some sparse results obtained for $k=1,2,3$. We establish estimates that optimally relate the higher order energies with the first order energy by carefully analyzing the effects of linear damping. The results concern weighted (in time) and also pointwise (in time…
We consider the problem of energy decay rates for nonlinearly damped abstract infinite dimensional systems. We prove sharp, simple and quasi-optimal energy decay rates through an indirect method, namely a weak observability estimate for the corresponding undamped system. One of the main advantage of these results is that they allow to combine the optimal-weight convexity method of Alabau-Boussouira and a methodology of Ammari-Tucsnak for weak stabilization by observability. Our results extend to nonlinearly damped systems, those of Ammari and Tucsnak. At the end, we give an appendix on the wea…
We study the decay rate of the energy of solutions to the damped wave equation in a setup where the geometric control condition is violated. We consider damping coefficients which are $0$ on a strip and vanish like polynomials, $x^β$. We prove that the semigroup cannot be stable at rate faster than $1/t^{(β+2)/(β+3)}$ by producing quasimodes of the associated stationary damped wave equation. We also prove that the semigroup is stable at rate at least as fast as $1/t^{(β+2)/(β+4)}$. These two results establish an explicit relation between the rate of vanishing of the damping and rate of decay o…
The aim of this paper is to prove a blow up result of the solution for a semilinear scale invariant damped wave equation under a suitable decay condition on radial initial data. The admissible range for the power of the nonlinear term depends both on the damping coefficient and on the pointwise decay order of the initial data. In addition we give an upper bound estimate for the lifespan of the solution, in terms of the power of the nonlinearity, size and growth of initial data.
In this paper, we establish a generalized H{ö}lder's or interpolation inequality for weighted spaces in which the weights are non-necessarily homogeneous. We apply it to the stabilization of some damped wave-like evolution equations. This allows obtaining explicit decay rates for smooth solutions for more general classes of damping operators. In particular, for $1-d$ models, we can give an explicit decay estimate for pointwise damping mechanisms supported on any strategic point.
We consider the wave equation with a damping term on a partially rectangular planar domain, assuming that the damping is concentrated close to the non-rectangular part of the domain. Polynomial decay estimates for the energy of the solution are established.
This work is devoted to improve the time decay estimates for the solution and some of its derivatives of the linear structurally damped $σ$-evolution equations. The Pitt inequality is the main tool provided that the initial data lies in some weighted spaces.
We revisit the damped wave equation on two-dimensional torus where the damped region does not satisfy the geometric control condition. We show that if the damping vanishes as a Hölder function $|x|^β$, and in addition, the boundary of the damped region is strictly convex, the wave is stable at rate $t^{-1+\frac{2}{2β+7}}$, which is better than the known optimal decay rate $t^{-1+\frac{1}{β+3}}$ for strip-shaped dampings of the same Hölder regularity. Moreover, we show by example that the decay rate is optimal. This illustrates the fact that the energy decay rate depends not only on the order o…
We study the energy decay rate of the Kelvin-Voigt damped wave equation with piecewise smooth damping on the multi-dimensional domain. Under suitable geometric assumptions on the support of the damping, we obtain the optimal polynomial decay rate which turns out to be different from the one-dimensional case studied in \cite{LR05}. This optimal decay rate is saturated by high energy quasi-modes localised on geometric optics rays which hit the interface along non orthogonal neither tangential directions. The proof uses semi-classical analysis of boundary value problems.
We consider the transmission problem for a coupled system of undamped and structurally damped plate equations in two sufficiently smooth and bounded subdomains. It is shown that, independently of the size of the damped part, the damping is strong enough to produce uniform exponential decay of the energy of the coupled system.
- In this article, we prove some universal bounds on the speed of concentration on small (frequency-dependent) neighborhoods of submanifolds of L 2-norms of quasi modes for Laplace operators on compact manifolds. We deduce new results on the rate of decay of weakly damped wave equations. R{é}sum{é}.
In this paper, we consider the energy decay of a damped hyperbolic system of wave-wave type which is coupled through the velocities. We are interested in the asymptotic properties of the solutions of this system in the case of indirect nonlinear damping, i.e. when only one equation is directly damped by a nonlinear damping. We prove that the total energy of the whole system decays as fast as the damped single equation. Moreover, we give a one-step general explicit decay formula for arbitrary nonlinearity. Our results shows that the damping properties are fully transferred from the damped equat…
The energy of solutions of the scalar damped wave equation decays uniformly exponentially fast when the geometric control condition is satisfied. A theorem of Lebeau [leb93] gives an expression of this exponential decay rate in terms of the average value of the damping terms along geodesics and of the spectrum of the infinitesimal generator of the equation. The aim of this text is to generalize this result in the setting of a vectorial damped wave equation on a Riemannian manifold with no boundary. We obtain an expression analogous to Lebeau's one but new phenomena like high frequency overdamp…
We prove local and global energy decay for the asymptotically periodic damped wave equation on the Euclidean space. Since the behavior of high frequencies is already mostly understood, this paper is mainly about the contribution of low frequencies. We show in particular that the damped wave behaves like a solution of a heat equation which depends on the H-limit of the metric and the mean value of the absorption index.
We establish the decay of the solutions of the damped wave equations in one dimensional space for the Dirichlet, Neumann, and dynamic boundary conditions where the damping coefficient is a function of space and time. The analysis is based on the study of the corresponding hyperbolic systems associated with the Riemann invariants. The key ingredient in the study of these systems is the use of the internal dissipation energy to estimate the difference of solutions with their mean values in an average sense.
In this paper, we analyze the pressureless damped Euler-Riesz equations posed in either $\mathbb{R}^d$ or $\mathbb{T}^d$. We construct the global-in-time existence and uniqueness of classical solutions for the system around a constant background state. We also establish large-time behaviors of classical solutions showing the solutions towards the equilibrium as time goes to infinity. For the whole space case, we first show the algebraic decay rate of solutions under additional assumptions on the initial data compared to the existence theory. We then refine the argument to have the exponential …
We consider the strongly damped Klein Gordon equation for defocusing nonlinearity and we study the asymptotic behaviour of the energy for periodic solutions. We prove first the exponential decay to zero for zero mean solutions. Then, we characterize the limit of the energy, when the time tends to infinity, for solutions with small enough initial data and we finally prove that such limit is not necessary zero.
In this paper, we discuss the global existence of weak solutions to the semilinear damped wave equation \begin{equation*} \begin{cases} \partial_t^2u-Δu + \partial_tu = f(u) & \text{in}\ Ω\times (0,T), \\ u=0 & \text{on}\ \partialΩ\times (0,T), \\ u(0)=u_0, \partial_tu(0)=u_1 & \text{in}\ Ω, \end{cases} \end{equation*} in an exterior domain $Ω$ in $\mathbb{R}^N$ $(N\geq 2)$, where $f:\mathbb{R}\to \mathbb{R}$ is a smooth function behaves like $f(u)\sim |u|^p$. From the view point of weighted energy estimates given by Sobajima--Wakasugi \cite{SoWa4}, the existence of global-in-time solutions wi…
We consider the damped wave equation on a manifold with imperfect geometric control. We show the sub-exponential energy decay estimate in \cite{Chr-NC-erratum} is optimal in the case of one hyperbolic periodic geodesic. We show if the equation is overdamped, then the energy decays exponentially. Finally we show if the equation is overdamped but geometric control fails for one hyperbolic periodic geodesic, then nevertheless the energy decays exponentially.
In this paper, we consider the Cauchy problem for semi-linear wave equations with structural damping term $ν(-Δ)^2 u_t$, where $ν>0$ is a constant. As being mentioned in [8,10], the linear principal part brings both the diffusion phenomenon and the regularity loss of solutions. This implies that, for the nonlinear problems, the choice of solution spaces plays an important role to obtain global solutions with sharp decay properties in time. Our main purpose of this paper is to prove the global (in time) existence of solutions for the small data and their decay properties for the supercritical n…
