Uniform Decay Research Articles

Stabilization of the wave equation with a nonlinear delay term in the boundary conditions

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.

Energy Decay Estimates of Solutions for Viscoelastic Damped Wave Equations in $$\mathbf {R}^{n}$$

In the present work, we investigate the global existence and uniform decay rates of solutions to the Cauchy problem in $$\mathbf {R}^{n}$$ ( $$n\ge 1)$$ related to the dynamic behavior of evolution equations accounting for rotational inertial forces along with a linear viscoelastic memory damping arising in viscoelastic materials. Under certain conditions on the initial data and on the kernel, the global existence and decay estimates of the solutions are established. Furthermore, time decay estimates in higher Sobolev space of the solution are provided. The proof is carried out by means of the point-wise decay estimates of the solution in the Fourier space.

Global Existence and Uniform Decay of Solutions for a Kirchhoff Beam Equation with Nonlinear Damping and Source Term

An extensible beam equation of Kirchhoff type with internal damping and source term is investigated. We apply the potential well and establish the global well-posedness of the initial and boundary value problem by using the Faedo-Galerkin approximations, taking into account that the initial data is located in a suitable set of stability created from the Nehari manifold. Moreover, by using Nakao’s method, we prove the exponential stability of the solution for $$p=1$$ and the polynomial stability for $$p>1$$ .

Existence and energy decay of solution to a nonlinear viscoelastic two-dimensional beam with a delay

A longitudinal and transversal vibrations of the beam with nonlinear tension, a viscoelastic damping and distributed delay term is studied. Using the Faedo–Galerkin method, the well-posedness of the problem is established. A uniform decay result is proved by multiplier method.

On the conservation of energy in two-dimensional incompressible flows

We prove the conservation of energy for weak and statistical solutions of the two-dimensional Euler equations, generated as strong (in an appropriate topology) limits of the underlying Navier–Stokes equations and a Monte Carlo-spectral viscosity numerical approximation, respectively. We characterize this conservation of energy in terms of a uniform decay of the so-called structure function, allowing us to extend existing results on energy conservation. Moreover, we present numerical experiments with a wide variety of initial data to validate our theory and to observe energy conservation in a large class of two-dimensional incompressible flows.

Linear modulational and subharmonic dynamics of spectrally stable Lugiato-Lefever periodic waves

We study the linear dynamics of spectrally stable T-periodic stationary solutions of the Lugiato-Lefever equation (LLE), a damped nonlinear Schrödinger equation with forcing that arises in nonlinear optics. Such T-periodic solutions are nonlinearly stable to NT-periodic, i.e. subharmonic, perturbations for each N∈N with exponential decay rates of perturbations of the form e−δNt. However, both the exponential rates of decay δN and the allowable size of the initial perturbations tend to 0 as N→∞, so that this result is non-uniform in N and, in fact, empty in the limit N=∞. The primary goal of this paper is to introduce a methodology, in the context of the LLE, by which a uniform stability result for subharmonic perturbations may be achieved, at least at the linear level. The obtained uniform decay rates are shown to agree precisely with the polynomial decay rates of localized, i.e. integrable on the real line, perturbations of such spectrally stable periodic solutions of the LLE. This work both unifies and expands on several existing works in the literature concerning the stability and dynamics of such waves, and sets forth a general methodology for studying such problems in other contexts.

Effects of domain unfolding and catch-like dissociation on the collective behavior of integrin–fibronectin bond clusters

To gain more understanding of cell–matrix adhesion, we consider an idealized theoretical model of a cluster of integrin–fibronectin bonds at the cell–matrix interface subjected to a dynamic ramping. The distributions of bond traction and interfacial deformation are assumed to obey classical elastic equations, whereas the dissociation/association of individual bonds as well as unfolding/refolding of fibronectin domains are described by stochastic equations. Through stochastic-elasticity coupling, we perform Monte Carlo simulations to investigate how the collective behavior and adhesion performance of the integrin–fibronectin-mediated interface are influenced by two characteristics newly incorporated in the modeling, i.e., catch-like dissociation between integrin and fibronectin, and unfolding of repeated domains in fibronectin. The probable unfolding of fibronectin domains is found to have profound effects on the resultant adhesion energy of the integrin–fibronectin-mediated interface, and governs the failure model transiting between uniform decay and catastrophic crack-like rupture. Focal adhesions containing integrin–fibronectin bond clusters play a vital role in a wide variety of cellular processes or functions. We develop a stochastic-elasticity modeling framework for such specific interaction at cell–matrix interface, by unifying the elastic descriptions of force distribution and deformation, and the stochastic descriptions of bond dissociation/association and domain unfolding/folding. The present modeling reveals a rich array of out-of-equilibrium behaviors of the integrin–fibronectin-mediated interface, and may provide clues for the development of artificial domain-containing structures and functional surfaces to modulate the collective behavior of bond-mediated cell–matrix adhesion.

Slow Decay of Waves in Gravitational Solitons

We consider a family of globally stationary (horizonless), asymptotically flat solutions of five-dimensional supergravity. We prove that massless linear scalar waves in such soliton spacetimes cannot have a uniform decay rate faster than inverse logarithmically in time. This slow decay can be attributed to the stable trapping of null geodesics. Our proof uses the construction of quasimodes which are time periodic approximate solutions to the wave equation. The proof is based on previous work to prove an analogous result in $$\text{ Kerr-AdS}_4$$ black holes (Holzegel and Smulevici in Anal PDE 7(5):1057–1090, 2014). We remark that this slow decay is suggestive of an instability at the nonlinear level.

A localization theorem for the planar Coulomb gas in an external field

We examine a Coulomb gas consisting of $n$ identical repelling point charges at an arbitrary inverse temperature $\beta$, subjected to a suitable external field. We prove that the gas is effectively localized to a small neighbourhood of the "droplet" -- the support of the equilibrium measure determined by the external field. More precisely, we prove that the distance between the droplet and the vacuum is with very high probability at most proportional to $$\sqrt{\dfrac {\log n}{\beta n}}.$$ This order of magnitude is known to be "tight" when $\beta=1$ and the external field is radially symmetric. In addition, we prove estimates for the one-point function in a neighbourhood of the droplet, proving in particular a fast uniform decay as one moves beyond a distance roughly of the order $\sqrt{\frac {\log n}{\beta n}}$ from the droplet.

Dynamical Behavior of the Indirectly and Locally Memory-Damped Timoshenko System

We are concerned with dynamical behavior of the indirectly and locally memory-damped Timoshenko system. The polynomial/exponential decay results for the long-term dynamical behavior of the Timoshenko system are established when the memory kernels decay polynomially/exponentially. As one can see, to obtain ideal asymptotic decay rates under basic conditions, we take some analysis processes specially designed for our issues. The obtained long-term dynamical behavior theorems, with the exact uniform decay rates for the solutions to the systems, indicate that for the Timoshenko system with indirect memory damping, a local memory effect is enough to produce an entire dissipation mechanism and to ensure the same decay rates as in the case of global memory effect. Moreover, although our conditions on the memory kernels are weaker than before, we still derive stronger conclusions than the former ones. Finally, we present some results of numerical simulation to illustrate quantitatively the behavior of the solution energies of the system under study, which verify our theoretical results well.

A geometric condition for the uniform stability of linear magnetoelasticity

We give a necessary and sufficient condition, of geometrical type, for the uniform decay of energy of solutions of the linear system of magnetoelasticity in a bounded domain with smooth boundary. A Dirichlet-type boundary condition is assumed. Our strategy is to use microlocal defect measures to show suitable observability inequalities on high-frequency solutions of the Lamé system.

On the viscoelastic equation with Balakrishnan-Taylor damping and nonlinear boundary/interior sources with variable-exponent nonlinearities

This work is devoted to the study of a nonlinear viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping and nonlinear boundary interior sources with variable exponents. Under appropriate assumptions, we establish a uniform decay rate of the solution energy in terms of the behavior of the nonlinear feedback and the relaxation function, without setting any restrictive growth assumptions on the damping at the origin and weakening the usual assumptions on the relaxation function.

Uniform decay rate estimates for the wave equation in an inhomogeneous medium with simultaneous interior and boundary feedbacks

Inspired by the highly cited work due to Lasiecka and Tataru (1993) [20], a semilinear model of the wave equation in an inhomogeneous medium with simultaneous interior and boundary feedbacks is considered. While for a homogeneous medium Ω and for certain geometries nonlinear feedback is strong enough to derive uniform decay rates of the energy associated with the model, in contrast, in an inhomogeneous medium, and for general geometry of Ω, the problem requires a new approach. In the first case, the rays of the geometric optics are straight lines and the geometry is favorable. However, in the second one, more damping is required to well-known obtain a similar result. For this purpose, arguments of microlocal analysis are used instead of the multipliers technique.

Uniform decay estimates for solutions of a class of retarded integral inequalities

Some uniform decay estimates are established for solutions of the following type of retarded integral inequalities:y(t)≤E(t,τ)‖yτ‖+∫τtK1(t,s)‖ys‖ds+∫t∞K2(t,s)‖ys‖ds+ρ,t≥τ≥0.As a simple example of application, the retarded scalar functional differential equation x˙=−a(t)x+B(t,xt) is considered, and the global asymptotic stability of the equation is proved under weaker conditions. Another example is the ODE system x˙=F0(t,x)+∑i=1mFi(t,x(t−ri(t))) on Rn with superlinear nonlinearities Fi (0≤i≤m). The existence of a global pullback attractor of the system is established under appropriate dissipation conditions.The third example for application concerns the study of the dynamics of the functional cocycle system dudt+Au=F(θtp,ut) in a Banach space X with sublinear nonlinearity. In particular, the existence and uniqueness of a nonautonomous equilibrium solution Γ is obtained under the hyperbolicity assumption on operator A and some additional hypotheses, and the global asymptotic stability of Γ is also addressed.

Exponential stability for infinite-dimensional non-autonomous port-Hamiltonian Systems

We study the non-autonomous version of an infinite-dimensional linear port-Hamiltonian system on an interval [ a , b ] . Employing abstract results on evolution families, we show C 1 -well-posedness of the corresponding Cauchy problem, and thereby existence and uniqueness of classical solutions for sufficiently regular initial data. Further, we demonstrate that a dissipation condition in the style of the dissipation condition sufficient for uniform exponential stability in the autonomous case also leads to a uniform exponential decay rate of the energy in this non-autonomous setting.

Long-term dynamical behavior of the wave model with locally distributed frictional and viscoelastic damping

We investigate the long-term dynamical behavior of the partially viscoelastic wave equation subject to a localized frictional damping, given byutt−Δu+∫0tg(t−s)div(a(x)∇u(s))ds+b(x)g1(ut)=0,inΩ×R+,where g denotes the memory kernel, a(x)∈C1(Ω¯),b(x)∈L∞(Ω) are nonnegative functions satisfying the assumptiona(x)+b(x)≥2δ>0,∀x∈ω0,ω0 is a subset of Ω, and b(x)g1(ut) denotes the frictional damping. Under as less as possible restrictions imposed on memory kernel g(·) and some geometric condition on the subset ω0, we show that there does not exist bifurcation and chaos for this physical model and actually the energy of the solution for the equation above decays definitely to zero with uniform decay rate as the time goes to infinity. Moreover, such a uniform decay rate is determined by the solution of an ordinary differential equation.

Decay rates for semilinear wave equations with vanishing damping and Neumann boundary conditions

The paper is concerned with the semilinear wave equations with time‐dependent damping γ(t)=α/(1+t) (α>0), under the effect of nonlinear source f behaving like a polynomial, and subject to Neumann boundary conditions. Constructing appropriate auxiliary functions, we obtain an explicit uniform decay rate estimate for the solutions of the equation in terms of the exponent of f, when α is large enough. On the other hand, via a new hyperbolic version of Dirichlet quotients, we show that the upper estimate is optimal in some case, which implies the existence of slow solutions.

Local solutions of the Landau equation with rough, slowly decaying initial data

We consider the Cauchy problem for the spatially inhomogeneous Landau equation with soft potentials in the case of large (i.e. non-perturbative) initial data. We construct a solution for any bounded, measurable initial data with uniform polynomial decay in the velocity variable, and that satisfies a technical lower bound assumption (but can have vacuum regions). For uniqueness in this weak class, we have to make the additional assumption that the initial data is Hölder continuous. Our hypotheses are much weaker, in terms of regularity and decay, than previous large-data well-posedness results in the literature. We also derive a continuation criterion for our solutions that is, for the case of very soft potentials, an improvement over the previous state of the art.

Global Existence and Decay Estimates of Solutions of a Parabolic–Elliptic–Parabolic System for Ion Transport Networks

This paper deals with a parabolic-elliptic-parabolic system arising from ion transport networks originally proposed by Albi et al. (Anal Appl 14(1):185–206, 2016). Our investigation demonstrates that the local existence of the weak solution can always be achieved. Moreover, by establishing the uniform a-priori decay estimates of solution, the local solution can be extended to a global one provided that the initial data is suitably small. In addition, the time-decay rate of the weak solution is obtained. To our best knowledge, this seems to be the first analytical work on this system indeed.

Uniform decay rate estimates for the coupled semilinear wave system in inhomogeneous media with locally distributed nonlinear damping

Uniform decay rate estimates for the coupled semilinear wave system in inhomogeneous media with locally distributed nonlinear damping