Exponential decay of eigenfunctions and generalized eigenfunctions of a non-self-adjoint matrix Schrodinger operator related to NLS

We study the decay of eigenfunctions of the non-self-adjoint, for µ > 0, corresponding to eigenvalues in the strip -µ < Re E < µ.     

Exponential decay of eigenfunctions in many-body type scattering with second-order perturbations

We show the exponential decay of eigenfunctions of second-order geometric many-body type Hamiltonians at non-threshold energies. Moreover, in the case of first-order and small second-order perturbations we show that there are no eigenfunctions with positive energy.     

Gaussian decay of the magnetic eigenfunctions

We investigate whether the eigenfunctions of the two-dimensional magnetic Schrödinger operator have a Gaussian decay of type exp(–Cx ²) at infinity (the magnetic field is rotationally symmetric). We establish this decay if the energy (E) of the eigenfunction is below the bottom of the essential spectrum (B), and if the angular Fourier components of the external potential decay exponentially (real analyticity in the angle variable). We also demonstrate that almost the same decay is necessary. The behavior ofC in the strong field limit and in the small (B–E) limit is also studied.Partial support from the Hungarian National Foundation for Scientific Research, grant no. 1902.     

Exponential decay for sc-gradient flow lines

In this paper we introduce the notion of sc-action functionals and their sc-gradient flow lines. Our approach is inspired by Floer’s unregularized gradient flow. The main result of this paper is that under a Morse condition, sc-gradient flow lines have uniform exponential decay towards critical points. The ultimate goal for the future is to construct an M-polyfold bundle over an M-polyfold such that the space of broken sc-gradient flow lines is the zero set of an appropriate sc-section. Here uniform exponential decay is essential. Of independent interest is that we derive exponential decay estimates using interpolation inequalities as opposed to Sobolev inequalities. An advantage is that interpolation inequalities are independent of the dimension of the source space.     

Exponential decay for the fragmentation or cell-division equation

We consider a classical integro-differential equation that arises in various applications as a model for cell-division or fragmentation. In biology, it describes the evolution of the density of cells that grow and divide. We prove the existence of a stable steady distribution (first positive eigenvector) under general assumptions in the variable coefficients case. We also prove the exponential convergence, for large times, of solutions toward such a steady state.     

Exponential decay of the solutions for nonlinear multiharmonic equations

The purpose of this paper is to establish the exponential decay properties of the solutions for the nonlinear multiharmonic equation

where the condition plays the role of a boundary value condition.     

Exponential decay estimates for singular integral operators

The following subexponential estimate for commutators is proved $$\begin{aligned} |\{x\in Q: |[b,T]f(x)|>tM^2f(x)\}|\le c\,e^{-\sqrt{\alpha \, t\Vert b\Vert _{BMO}}}\, |Q|, \qquad t>0. \end{aligned}$$ where $c$ and $\alpha $ are absolute constants, $T$ is a Calderón–Zygmund operator, $M$ is the Hardy Littlewood maximal function and $f$ is any function supported on the cube $Q\subset \mathbb{R }^n$ . We also obtain that $$\begin{aligned} |\{x\in Q: |f(x)-m_f(Q)|>tM_{\lambda _n;Q}^\#(f)(x) \}|\le c\, e^{-\alpha \,t}|Q|,\qquad t>0, \end{aligned}$$ where $m_f(Q)$ is the median value of $f$ on the cube $Q$ and $M_{\lambda _n;Q}^\#$ is Strömberg’s local sharp maximal function with $\lambda _n=2^{-n-2}$ . As a consequence we derive Karagulyan’s estimate: $$\begin{aligned} |\{x\in Q: |Tf(x)|> tMf(x)\}|\le c\, e^{-c\, t}\,|Q|\qquad t>0, \end{aligned}$$ from [21] improving Buckley’s theorem [3]. A completely different approach is used based on a combination of “Lerner’s formula” with some special weighted estimates of Coifman–Fefferman type obtained via Rubio de Francia’s algorithm. The method is flexible enough to derive similar estimates for other operators such as multilinear Calderón–Zygmund operators, dyadic and continuous square functions and vector valued extensions of both maximal functions and Calderón–Zygmund operators. In each case, $M$ will be replaced by a suitable maximal operator.     

Exponential decay of expansive constants

A map f on a compact metric space is expansive if and only if fn is expansive.We study the exponential rate of decay of the expansive constant of fn and find some of its relations with other quantities about the dynamics,such as box dimension and topological entropy.     

Exponential decay for a viscoelastically damped timoshenko beam

Of concern is a viscoelastic beam modelled using the Timoshenko theory. It is well-known that the system is exponentially stable if the kernel in the memory term is sub-exponential. That is, if the product of the kernel with an exponential function is a summable function. In this article we address the questions: What if the kernel is tested against a different function (say Gamma) other than the exponential function? Would there still be stability? In the affirmative, what kind of decay rate we get? It is proved that for a non-decreasing function “Gamma” whose “logarithmic derivative” is decreasing to zero we have a decay of order Gamma to some power and in the case it decreases to a different value than zero then the decay is exponential.     

Exponential decay for semilinear distributed systems with damping

The stability of second order abstract distributed systems with damping and nonlinear perturbations is considered. Sufficient conditions, including unique continuation property assumptions, are formulated to obtain (local, non-uniform and uniform) exponential stability. Applications to the wave and Euler-Bernoulli equations are given.     

Criteria of exponential decay for eigenfunctions of some classes of integral operators

We study sufficient conditions for exponential decay at infinity for eigenfunctions of a class of integral equations in unbounded domains in ℝ ⁿ . We consider integral operators K whose kernels have the form

Exponential decay to a quantum hydrodynamic model for semiconductors

This paper deals with large-time behavior of solutions for a viscous bipolar quantum hydrodynamic model with third-order terms. By applying the entropy method, we prove exponential decays of solutions towards constant steady states for the one-dimensional and the multi-dimensional cases. The argument is based on a series of a priori estimates. As a byproduct, the decay of solutions for the viscous hydrodynamic model is obtained as well.     

Exponential decay for distributed bilinear control systems with damping

Let Ω be a bounded open domain in ℝ ^N ,A an unbounded, selfadjoint, positive and coercive linear operator onL ² (Ω). We consider feedback stabilization for the distributed bilinear control systemy″(t)+Ay(t)+Dy′(t)+u(t)By(t)=0, whereD andB are linear bounded operators fromL ²(Ω) toL ²(Ω). Under suitable assumptions onB andD, a nonlinear feedback ensuring uniform exponential decay of solutions is given. Various applications to vibrating processes are presented.     

Exponential decay for the one-dimensional wave equation with internal pointwise damping

The paper considers a particular type of closed-loop for the wave equation in one space dimension with damping acting at an arbitrary internal point, for which the uniform stabilization with exponential decay rate is shown. Applications to chains of coupled strings are also discussed. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Exponential decay for a viscoelastic problem with a singular kernel

In this paper we consider a problem which arises in viscoelasticity. We prove exponential decay of solutions for the problem with a memory term involving a kernel which is singular at zero. This is established by introducing an appropriate Lyapunov type functional and using the energy method. This work extends earlier results.     

Exponential decay for a viscoelastic problem with a singular kernel

In this paper we consider a problem which arises in viscoelasticity. We prove exponential decay of solutions for the problem with a memory term involving a kernel which is singular at zero. This is established by introducing an appropriate Lyapunov type functional and using the energy method. This work extends earlier results.      

Exponential decay for Maxwell equations with a boundary memory condition

We study the asymptotic behavior of the solution of the Maxwell equations with the following boundary condition of memory type:

(0.1)