In this paper we present new results on the preservation of polynomial stability of damped wave equations under addition of perturbing terms. We in particular introduce sufficient conditions for the stability of perturbed two-dimensional wave equations on rectangular domains, a one-dimensional weakly damped Webster's equation, and a wave equation with an acoustic boundary condition. In the case of Webster's equation, we use our results to compute explicit numerical bounds that guarantee the polynomial stability of the perturbed equation.
We consider a class of wave equations of the type $\partial_{tt} u + Lu + B\partial_{t} u = 0$, with a self-adjoint operator $L$, and various types of local damping represented by $B$. By establishing appropriate and raher precise estimates on the resolvent of an associated operator $A$ on the imaginary axis of ${\Bbb C}$, we prove polynomial decay of the semigroup $\exp(-tA)$ generated by that operator. We point out that the rate of decay depends strongly on the concentration of eigenvalues and that of the eigenfunctions of the operator $L$. We give several examples of application of our abst…
We study decay rates for the energy of solutions of the damped wave equation on the torus. We consider dampings invariant in one direction and bounded above and below by multiples of $x^β$ near the boundary of the support and show decay at rate $1/t^{\frac{β+2}{β+3}}$. In the case where $W$ vanishes exactly like $x^β$ this result is optimal by work of the second author. The proof uses a version of the Morawetz multiplier method.
We study the decay of global energy for the wave equation with Hölder continuous damping placed on the $C^{1,1}$-boundary of compact and non-compact waveguides with star-shaped cross-sections. We show there is sharp $t^{-1/2}$-decay when the damping is uniformly bounded from below on the cylindrical wall of product cylinders where the Geometric Control Condition is violated. On non-product cylinders, we also show that there is $t^{-1/3}$-decay when the damping is uniformly bounded from below on the cylindrical wall.
We consider a transmission problem where a structurally damped plate equation is coupled with a damped or undamped wave equation by transmission conditions. We show that exponential stability holds in the damped-damped situation and polynomial stability (but no exponential stability) holds in the damped-undamped case. Additionally, we show that the solutions first defined by the weak formulation, in fact have higher Sobolev space regularity.
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.
In this paper, by means of the Riesz basis approach, we study the stability of a weakly damped system of two second order evolution equations coupled through the velocities. If the fractional order damping becomes viscous and the waves propagate with equal speeds, we prove exponential stability of the system and, otherwise, we establish an optimal polynomial decay rate. Finally, we provide some illustrative examples.
In this paper we show how to obtain decay estimates for the damped wave equation on a compact manifold without geometric control via knowledge of the dynamics near the un-damped set. We show that if replacing the damping term with a higher-order \emph{complex absorbing potential} gives an operator enjoying polynomial resolvent bounds on the real axis, then the "resolvent" associated to our damped problem enjoys bounds of the same order. It is known that the necessary estimates with complex absorbing potential can also be obtained via gluing from estimates for corresponding non-compact models.
A strongly damped wave equation including the displacement depending nonlinear damping term and nonlinear interaction function is considered. The main aim of the note is to show that under the standard dissipativity restrictions on the nonlinearities involved the initial boundary value problem for the considered equation is globally well-posed in the class of sufficiently regular solutions and the semigroup generated by the problem possesses a global attractor in the corresponding phase space. These results are obtained for the nonlinearities of an arbitrary polynomial growth and without the a…
We study the so-called damped Navier-Stokes equations in the whole 2D space. The global well-posedness, dissipativity and further regularity of weak solutions of this problem in the uniformly-local spaces are verified based on the further development of the weighted energy theory for the Navier-Stokes type problems. Note that any divergent free vector field $u_0\in L^\infty(\mathbb R^2)$ is allowed and no assumptions on the spatial decay of solutions as $|x|\to\infty$ are posed. In addition, applying the developed theory to the case of the classical Navier-Stokes problem in R2, we show that th…
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 demonstrate the global existence of weak solutions to a class of semilinear strongly damped wave equations possessing nonlinear hyperbolic dynamic boundary conditions. Our work assumes $(-Δ_W)^θ\partial_tu$ with $θ\in[\frac{1}{2},1)$ and where $Δ_W$ is the Wentzell-Laplacian. Hence, the associated linear operator admits a compact resolvent. A balance condition is assumed to hold between the nonlinearity defined on the interior of the domain and the nonlinearity on the boundary. This allows for arbitrary (supercritical) polynomial growth on each potential, as well as mixed dissipative/anti-d…
We study a damped stochastic non-linear Schrödinger (NLS) equation driven by an additive noise. It is white in time and smooth in space. Using a coupling method, we establish convergence of the Markovian transition semi-group toward a unique invariant probability measure. This kind of method was originally developped to prove exponential mixing for strongly dissipative equations such as the Navier-Stokes equations. We consider here a weakly dissipative equation, the damped nonlinear Schrödinger equation in the one dimensional cubic case. We prove that the mixing property holds and that the rat…
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 open question, which seems to be also the final part, in terms of studying the Cauchy problem for the weakly coupled system of damped wave equations or reaction-diffusion equations, is so far known as the sharp lifespan estimates in the critical case. In this paper, we mainly investigate lifespan estimates for solutions to the weakly coupled system of semilinear damped wave equations in the critical case. By using a suitable test function method associated with nonlinear differential inequalities, we catch upper bound estimates for the lifespan. Moreover, we establish polynomial-logarithmi…
We prove local energy decay for the damped wave equation on R^d. The problem which we consider is given by a long range metric perturbation of the Euclidean Laplacian with a short range absorption index. Under a geometric control assumption on the dissipation we obtain an almost optimal polynomial decay for the energy in suitable weighted spaces. The proof relies on uniform estimates for the corresponding "resolvent", both for low and high frequencies. These estimates are given by an improved dissipative version of Mourre's commutators method.
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…
In this paper, we investigate the direct and indirect stability of locally coupled wave equations with local viscous damping on cylindrical and non-regular domains without any geometric control condition. If only one equation is damped, we prove that the energy of our system decays polynomially with the rate $t^{-\frac{1}{2}}$ if the two waves have the same speed of propagation, and with rate $t^{-\frac{1}{3}}$ if the two waves do not propagate at the same speed. Otherwise, in case of two damped equations, we prove a polynomial energy decay rate of order $t^{-1}$.
We present a new method of investigating the so-called quasi-linear strongly damped wave equations $$ \partial_t^2u-γ\partial_tΔ_x u-Δ_x u+f(u)= \nabla_x\cdot ϕ'(\nabla_x u)+g $$ in bounded 3D domains. This method allows us to establish the existence and uniqueness of energy solutions in the case where the growth exponent of the non-linearity $ϕ$ is less than 6 and $f$ may have arbitrary polynomial growth rate. Moreover, the existence of a finite-dimensional global and exponential attractors for the solution semigroup associated with that equation and their additional regularity are also estab…
Let $u$ solve the damped Klein--Gordon equation $$ \big( \partial_t^2-\sum \partial_{x_j}^2 +m \text{Id} +γ(x) \partial_t \big) u=0 $$ on $\mathbb{R}^n$ with $m>0$ and $γ\geq 0$ bounded below on a $2 π\mathbb{Z}^n$-invariant open set by a positive constant. We show that the energy of the solution $u$ decays at a polynomial rate. This is proved via a periodic observability estimate on $\mathbb{R}^n.$
Stability and Finite-Time Blow-Up for a Fractionally Damped Nonlinear Plate Equation: Numerical and Analytical Insights
School of Mathematics and Statistics, Shanxi University（山西大学数学与统计学院）；Department of Mathematics, University of Kotli Azad Jammu and Kashmir(UOKAJK)（阿扎德克什米尔kotli大学数学系）；DIM CIMA Universidad de Concepción（康塞普西翁大学DIM CIMA）
We rigorously derive multi-pulse interaction laws for the semi-strong interactions in a family of singularly-perturbed and weakly-damped reaction-diffusion systems in one space dimension. Most significantly, we show the existence of a manifold of quasi-steady N-pulse solutions and identify a "normal-hyperbolicity" condition which balances the asymptotic weakness of the linear damping against the algebraic evolution rate of the multi-pulses. Our main result is the adiabatic stability of the manifolds subject to this normal hyperbolicity condition. More specifically, the spectrum of the lineariz…
We establish upper bounds for the decay rate of the energy of the damped fractional wave equation when the averages of the damping coefficient on all intervals of a fixed length are bounded below. If the power of the fractional Laplacian, $s$, is between 0 and 2, the decay is polynomial. For $s \ge 2$, the decay is exponential. Second, we show that our assumption on the damping is necessary for the energy to decay exponentially.
On generalised d'Alembert-type integral representations for damped wave equations on the quarter-plane
National and Kapodistrian University of Athens（雅典国立卡波蒂斯特里安大学）；University of Cyprus（塞浦路斯大学）；Universita degli Studi di Brescia（布雷西亚大学）；Istituto Nazionale di Fisica Nucleare, Milano Bicocca（意大利国家核物理研究所，米兰比科卡）；Henan Normal University（河南师范大学）；Laboratoire de Mathématiques Appliquées du Havre (LMAH), Université Le Havre Normandie（勒阿弗尔应用数学实验室，勒阿弗尔诺曼底大学）
In this paper, we examine regularity and stability issues for two damped abstract elastic systems. The damping involves the average velocity and a fractional power $θ$, with $θ$ in $[-1,1]$, of the principal operator. The matrix operator defining the damping mechanism for the coupled system is degenerate. First, we prove that for $θ$ in $(1/2,1]$, the underlying semigroup is not analytic, but is differentiable for $θ$ in $(0,1)$; this is in sharp contrast with known results for a single similarly damped elastic system, where the semigroup is analytic for $θ$ in $[1/2,1]$; this shows that the d…
In this paper, we investigate the energy decay of hyperbolic systems of wave-wave, wave-Euler- Bernoulli beam and beam-beam types. The two equations are coupled through boundary connection with only one localized non-smooth fractional Kelvin-Voigt damping. First, we reformulate each system into an augmented model and using a general criteria of Arendt-Batty, we prove that our models are strongly stable. Next, by using frequency domain approach, combined with multiplier technique and some interpolation inequalities, we establish different types of polynomial energy decay rate which depends on t…
