Uniform Decay Rates Research Articles

Rapid Transcription Fosters Coordinate snail Expression in the Drosophila Embryo

Transcription is commonly held to be a highly stochastic process, resulting in considerable heterogeneity of gene expression among the different cells in a population. Here, we employ quantitative in situ hybridization methods coupled with high-resolution imaging assays to measure the expression of snail, a developmental patterning gene necessary for coordinating the invagination of the mesoderm during gastrulation of the Drosophila embryo. Our measurements of steady-state mRNAs suggest that there is very little variation in snail expression across the different cells that make up the mesoderm and that synthesis approaches the kinetic limits of Pol II processivity. We propose that rapid transcription kinetics and negative autoregulation are responsible for the remarkable homogeneity of snail expression and the coordination of mesoderm invagination.

Global existence and decay estimates for nonlinear Kirchhoff-type equation with boundary dissipation

In this paper, we consider the initial-boundary value problem for nonlinear Kirchhofftype equation utt −φ(‖∇u‖2)Δu−aΔut = b|u|β−2u, where a,b > 0 and β > 2 are constants, φ is a C1 -function such that φ(s) λ0 > 0 for all s 0 . Under suitable conditions on the initial data, we show the existence and uniqueness of global solution by means of the Galerkin method and the uniform decay rate of the energy by an integral inequality. Mathematics subject classification (2010): 35J05.

PT-Symmetric Quantum Liouvillean Dynamics

We discuss a combination of unitary and antiunitary symmetry of quantum Liouvillean dynamics, in the context of open quantum systems, which implies a D2 symmetry of the complex Liouvillean spectrum. For sufficiently weak system-bath coupling, it implies a uniform decay rate for all coherences, i.e., off-diagonal elements of the system's density matrix taken in the eigenbasis of the Hamiltonian. As an example, we discuss symmetrically boundary driven open XXZ spin 1/2 chains.

Global existence and decay of energy to systems of wave equations with damping and supercritical sources

This paper is concerned with a system of nonlinear wave equations with supercritical interior and boundary sources and subject to interior and boundary damping terms. It is well-known that the presence of a nonlinear boundary source causes significant difficulties since the linear Neumann problem for the single wave equation is not, in general, well-posed in the finite-energy space H1(Ω) × L2(∂Ω) with boundary data from L2(∂Ω) (due to the failure of the uniform Lopatinskii condition). Additional challenges stem from the fact that the sources considered in this article are non-dissipative and are not locally Lipschitz from H1(Ω) into L2(Ω) or L2(∂Ω). With some restrictions on the parameters in the system and with careful analysis involving the Nehari Manifold, we obtain global existence of a unique weak solution and establish (depending on the behavior of the dissipation in the system) exponential and algebraic uniform decay rates of energy. Moreover, we prove a blow-up result for weak solutions with nonnegative initial energy.

Geometrically constrained stabilization of wave equations with Wentzell boundary conditions

Uniform stabilization of wave equation subject to second-order boundary conditions is considered in this article. Both dynamic (Wentzell) and static (with higher derivatives in space only) boundary conditions are discussed. In contrast to the classical wave equation where stabilization can be achieved by applying boundary velocity feedback, for a Wentzell-type problem boundary damping alone does not cause the energy to decay uniformly to zero. This is the case for both dynamic and static second-order conditions. In order to achieve uniform decay rates of the associated energy, it is necessary to dissipate part of the collar near the boundary. It will be shown how a combination of partially localized boundary feedback and partially localized collar feedback leads to uniform decay rates that are described by a nonlinear differential equation. This goal is attained by combining techniques used for stabilization of ‘unobserved’ Neumann conditions with differential geometry techniques effective for stabilization on compact manifolds. These lead to a construction of special non-radial multipliers which are geometry dependent and allow reconstruction of the high-order part of the potential energy from the damping that is supported only in a far-off region of the domain.

Price’s law on nonstationary space–times

In this article, we study the pointwise decay properties of solutions to the wave equation on a class of nonstationary asymptotically flat backgrounds in three space dimensions. Under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establish a t−3 local uniform decay rate (Price’s law, Price (1972) [54]) for linear waves. As a corollary, we also prove Price’s law for certain small perturbations of the Kerr metric.This result was previously established by the second author in (Tataru [65]) on stationary backgrounds. The present work was motivated by the problem of nonlinear stability of the Kerr/Schwarzschild solutions for the vacuum Einstein equations, which seems to require a more robust approach to proving linear decay estimates.

On Existence, Uniform Decay Rates, and Blow‐Up for Solutions of a Nonlinear Wave Equation with Dissipative and Source

This paper studies the blow‐up and existence, and asymptotic behaviors of the solution of a nonlinear hyperbolic equation with dissipative and source terms. By using Galerkin procedure and the perturbed energy method, the local and global existence of solution is established. In addition, by the concave method, the blow‐up of solutions can be obtained.

Uniform boundary stabilization of a wave equation with nonlinear acoustic boundary conditions and nonlinear boundary damping

We consider a wave equation with nonlinear acoustic boundary conditions. This is a nonlinearly coupled system of hyperbolic equations modeling an acoustic/structure interaction, with an additional boundary damping term to induce both existence of solutions as well as stability. Using the methods of Lasiecka and Tataru for a wave equation with nonlinear boundary damping, we demonstrate well-posedness and uniform decay rates for solutions in the finite energy space, with the results depending on the relationship between (i) the mass of the structure, (ii) the nonlinear coupling term, and (iii) the size of the nonlinear damping. We also show that solutions (in the linear case) depend continuously on the mass of the structure as it tends to zero, which provides rigorous justification for studying the case where the mass is equal to zero.

General decay and blow-up of solutions for a viscoelastic equation with nonlinear boundary damping-source interactions

In this paper, a viscoelastic equation with nonlinear boundary damping and source terms of the form $$\begin{array}{llll}u_{tt}(t)-\Delta u(t)+\displaystyle\int\limits_{0}^{t}g(t-s)\Delta u(s){\rm d}s=a\left\vert u\right\vert^{p-1}u,\quad{\rm in}\,\Omega\times(0,\infty), \\ \qquad\qquad\qquad\qquad\qquad u=0,\,{\rm on}\,\Gamma_{0} \times(0,\infty),\\ \dfrac{\partial u}{\partial\nu}-\displaystyle\int\limits_{0}^{t}g(t-s)\frac{\partial}{\partial\nu}u(s){\rm d}s+h(u_{t})=b\left\vert u\right\vert ^{k-1}u,\quad{\rm on} \ \Gamma_{1} \times(0,\infty) \\ \qquad\qquad\qquad\qquad u(0)=u^{0},u_{t}(0)=u^{1},\quad x\in\Omega, \end{array}$$ is considered in a bounded domain Ω. Under appropriate assumptions imposed on the source and the damping, we establish both existence of solutions and uniform decay rate of the solution energy in terms of the behavior of the nonlinear feedback and the relaxation function g, without setting any restrictive growth assumptions on the damping at the origin and weakening the usual assumptions on the relaxation function g. Moreover, for certain initial data in the unstable set, the finite time blow-up phenomenon is exhibited.

On a wave equation with supercritical interior and boundary sources and damping terms

This article addresses nonlinear wave equations with supercritical interior and boundary sources, and subject to interior and boundary damping. The presence of a nonlinear boundary source alone is known to pose a significant difficulty since the linear Neumann problem for the wave equation is not, in general, well-posed in the finite-energy space H1(Ω) × L2(∂Ω) with boundary data in L2 due to the failure of the uniform Lopatinskii condition. Further challenges stem from the fact that both sources are non-dissipative and are not locally Lipschitz operators from H1(Ω) into L2(Ω), or L2(∂Ω). With some restrictions on the parameters in the model and with careful analysis involving the Nehari Manifold, we obtain global existence of a unique weak solution, and establish exponential and algebraic uniform decay rates of the finite energy (depending on the behavior of the dissipation terms). Moreover, we prove a blow up result for weak solutions with nonnegative initial energy.

Periodic solutions for a weakly dissipated hybrid system

We consider the motion of a stretched string coupled with a rigid body at one end and we study the existence of periodic solution when a periodic force f acts on the body. The main difficulty of the study is related to the weak dissipation that characterizes this hybrid system, which does not ensure a uniform decay rate of the energy. Under additional regularity conditions on f, we use a perturbation argument in order to prove the existence of a periodic solution. In the last part of the paper we present some numerical simulations based on the theoretical results.

Uniform decay rates of coupled anisotropic elastodynamic/Maxwell equations with nonlinear damping

This work is devoted to study the asymptotic behavior of the total energy associated with a coupled system of anisotropic hyperbolic models: the elastodynamic equations and Maxwell's system in the exterior of a bounded body in \mathbb{R}^3 . Our main result says that in the presence of nonlinear damping, a unique solution of small initial data exists globally in time and the total energy as well as higher order energies decay at a uniform rate as t \rightarrow + \infty .

Limit-periodic Schrödinger operators with uniformly localized eigenfunctions

We exhibit limit-periodic Schrödinger operators that are uniformly localized in the strongest sense possible. That is, for these operators there are uniform exponential decay rates such that every element of the hull has a complete set of eigenvectors that decay exponentially off their centers of localization at least as fast as prescribed by the uniform decay rate. Consequently, these operators exhibit uniform dynamical localization.

Uniform decay rates for the energy of weakly damped defocusing semilinear Schrödinger equations with inhomogeneous Dirichlet boundary control

In this paper, we study the open loop stabilization as well as the existence and regularity of solutions of the weakly damped defocusing semilinear Schrödinger equation with an inhomogeneous Dirichlet boundary control. First of all, we prove the global existence of weak solutions at the H 1 -energy level together with the stabilization in the same sense. It is then deduced that the decay rate of the boundary data controls the decay rate of the solutions up to an exponential rate. Secondly, we prove some regularity and stabilization results for the strong solutions in H 2 -sense. The proof uses the direct multiplier method combined with monotonicity and compactness techniques. The result for weak solutions is strong in the sense that it is independent of the dimension of the domain, the power of the nonlinearity, and the smallness of the initial data. However, the regularity and stabilization of strong solutions are obtained only in low dimensions with small initial and boundary data.

Uniform decay rate for the transmission wave equations with variable coefficients

This paper considers the stabilization of the transmission problem of wave equations with variable coefficients. By introducing both boundary feedback control and distribute feedback control near the transmission boundary, the author establishes the uniform energy decay rate for the problem.

Uniform energy decay rates for nonlinear viscoelastic wave equation with nonlocal boundary damping

In this paper, we consider the uniform decay estimates of solutions for the viscoelastic wave equation u t t − κ 0 Δ u + ∫ 0 t g ( t − s ) div [ a ( x ) ∇ u ( s ) ] d s + b ( x ) h ( u t ) = 0 in Ω × ( 0 , ∞ ) . Under weak assumptions on the functions g , h and f , we prove the energy functional decays exponentially or polynomially to zero as the time goes to infinity by introducing brief Lyapunov functions and precise priori estimates.

Global existence uniqueness and decay estimates for nonlinear viscoelastic wave equation with boundary dissipation

In this paper, we consider the nonlinear viscoelastic wave equation u t t − Δ u + ∫ 0 t h ( t − τ ) div [ a ( x ) ∇ u ( τ ) ] d τ + | u | γ u = 0 with boundary dissipation. Under suitable conditions on the initial data and the relaxation function, we prove the existence and uniqueness of its global solution by means of the Galerkin method and show the uniform decay rate of the energy.

Global existence and uniform decay of solutions for a system of wave equations with dispersive and dissipative terms

In this paper, we consider a system of two coupled wave equations with dispersive and viscosity dissipative terms under Dirichlet boundary conditions. The global existence of weak solutions as well as uniform decay rates (exponential one) of the solution energy are established.

Exponential stability of time-varying linear systems

This paper considers the stability of both continuous and discrete time-varying linear systems. Stability estimates are obtained in either case in terms of the Lipschitz constant for the governing matrices and the assumed uniform decay rate of the corresponding frozen time linear systems. The main techniques used in the analysis are comparison methods, scaling and the application of continuous stability estimates to the discrete case. Counterexamples are presented to show the necessity of the stability hypotheses. The discrete results are applied to derive sufficient conditions for the stability of a backward Euler approximation of a time-varying system and a one-leg linear multistep approximation of a scalar system.

Rate of decay of solutions of the wave equation with arbitrary localized nonlinear damping

We study the rate of decay of solutions of the wave equation with localized nonlinear damping without any growth restriction and without any assumption on the dynamics. Providing regular initial data, the asymptotic decay rates of the energy functional are obtained by solving nonlinear ODE. Moreover, we give explicit uniform decay rates of the energy. More precisely, we find that the energy decays uniformly at last, as fast as 1 / ( ln ( t + 2 ) ) 2 − δ , ∀ δ > 0 , when the damping has a polynomial growth or sublinear, and for an exponential damping at the origin the energy decays at last, as fast as 1 / ( ln ( ln ( t + e 2 ) ) ) 2 − δ , ∀ δ > 0 .