Monotonicity of stable solutions in shadow systems

A shadow system appears as a limit of a reaction-diffusion system in which some components have infinite diffusivity. We investigate the spatial structure of its stable solutions. It is known that, unlike scalar reaction-diffusion equations, some shadow systems may have stable nonconstant (monotone) solutions. On the other hand, it is also known that in autonomous shadow systems any nonconstant non-monotone stationary solution is necessarily unstable. In this paper, it is shown in a general setting that any stable bounded (not necessarily stationary) solution is asymptotically homogeneous or eventually monotone in .

     

Stability switches in a system of linear differential equations with diagonal delay

This paper deals with the stability problem of a delay differential system of the form x^′(t)=-ax(t-τ)-by(t), y^′(t)=-cx(t)-ay(t-τ), where a, b, and c are real numbers and τ is a positive number. We establish some necessary and sufficient conditions for the zero solution of the system to be asymptotically stable. In particular, as τ increases monotonously from 0, the zero solution of the system switches finite times from stability to instability to stability if ; and from instability to stability to instability if . As an application, we investigate the local asymptotic stability of a positive equilibrium of delayed Lotka-Volterra systems.     

Lack of exponential stability to Timoshenko system with viscoelastic Kelvin–Voigt type

We study the Timoshenko systems with a viscoelastic dissipative mechanism of Kelvin–Voigt type. We prove that the model is analytical if and only if the viscoelastic damping is present in both the shear stress and the bending moment. Otherwise, the corresponding semigroup is not exponentially stable no matter the choice of the coefficients. This result is different to all others related to Timoshenko model with partial dissipation, which establish that the system is exponentially stable if and only if the wave speeds are equal. Finally, we show that the solution decays polynomially to zero as \({t^{-1/2}}\) , no matter where the viscoelastic mechanism is effective and that the rate is optimal whenever the initial data are taken on the domain of the infinitesimal operator.     

A note on asymptotically flat metrics on which are scalar-flat and admit minimal spheres

We use constructions by Miao and Chrusciel-Delay to produce asymptotically flat metrics on which have zero scalar curvature and multiple stable minimal spheres. Such metrics are solutions of the time-symmetric vacuum constraint equations of general relativity, and in this context the horizons of black holes are stable minimal spheres. We also note that under pointwise sectional curvature bounds, asymptotically flat metrics of nonnegative scalar curvature and small mass do not admit minimal spheres, and hence are topologically .

     

Stability criteria for linear second order delay differential system

Consider second order delay differential system where r is a positive constant and all coefficients are real constants.Our main results are as follows:(1) The maximal length of the delay for which the stability of system (*) is maintained is given in the case where the zero solution of system (*) is asymptotically stable in the absence of delay.(2) The necessary and sufficient criteria for judging that asymptetical stability of system (*) is preserved for an arbitrary large delay are obtained.     

Existence and uniqueness for spatially inhomogeneous coagulation equation with sources and effluxes

We prove the local existence theorem for general Smoluchovsky's coagulation equation with coagulation kernels which allow the multiplicative growth. If the system concerned has absorption, then the local existence theorem converts into the global existence theorem provided that initial data and sources are sufficiently small. We prove uniqueness, mass conservation and continuous dependence on initial data in the domain of its existence. We show that the solution in large asymptotically tends to zero as time goes to infinity and demonstrate that, in general, the sequence of approximated solutions does not converge to the exact solution of the original problem with the multiplicative kernel. This fact reveals the limits of numerical simulation of the coagulation equation.     

Pointwise optimality of Bayesian wavelet estimators

We consider pointwise mean squared errors of several known Bayesian wavelet estimators, namely, posterior mean, posterior median and Bayes Factor, where the prior imposed on wavelet coefficients is a mixture of an atom of probability zero and a Gaussian density. We show that for the properly chosen hyperparameters of the prior, all the three estimators are (up to a log-factor) asymptotically minimax within any prescribed Besov ball . We discuss the Bayesian paradox and compare the results for the pointwise squared risk with those for the global mean squared error.     

STABILITY AND EXISTENCE OF PERIODIC SOLUTIONS AND ALMOST PERIODIC SOLUTIONS ON LIENARD SYSTEMS

1MainResultsConsidersystem11~.x f(x)x' g(x)~0(1)wheref(x)islocallyintegrable,g(x)isdifferentiablealldg(0)=0.Theroem1Thezerosolutionofsystem(1)isuniformlyasymptoticallystableifbyequivalenttransf'Ormu=xov=X' F(x).DefineW[t,(uif\v)]j6ug(s)ds Iv',thenwisaposi…     

Asymptotic analysis of solutions to parabolic systems

We study asymptotics as of solutions to a linear, parabolic system of equations with time-dependent coefficients in , where is a bounded domain. On we prescribe the homogeneous Dirichlet boundary condition. For large values of t, the coefficients in the elliptic part are close to time-independent coefficients in an integral sense which is described by a certain function . This includes in particular situations when the coefficients may take different values on different parts of and the boundaries between them can move with t but stabilize as . The main result is an asymptotic representation of solutions for large t. A consequence is that for , the solution behaves asymptotically as the solution to a parabolic system with time-independent coefficients.     

Products of polynomials in uniform norms

We study inequalities connecting a product of uniform norms of polynomials with the norm of their product. This subject includes the well known Gel'fond-Mahler inequalities for the unit disk and Kneser inequality for the segment . Using tools of complex analysis and potential theory, we prove a sharp inequality for norms of products of algebraic polynomials over an arbitrary compact set of positive logarithmic capacity in the complex plane. The above classical results are contained in our theorem as special cases.

It is shown that the asymptotically extremal sequences of polynomials, for which this inequality becomes an asymptotic equality, are characterized by their asymptotically uniform zero distributions. We also relate asymptotically extremal polynomials to the classical polynomials with asymptotically minimal norms.

     

Stability and almost periodicity of solutions of ill-posed abstract Cauchy problems

We give simple spectral sufficient conditions for a solution of the linear abstract Cauchy problem, on a Banach space, to be strongly stable or asymptotically almost periodic, without assuming that the associated operator generates a -semigroup.

     

Time-periodic Ginzburg-Landau equations for one dimensional patterns with large wave length

Summary We consider a system invariant under shifts in the spatial coordinatex, and under reflection symmetryx –x, and possessing a fully symmetric steady solution. We assume that instability threshold of this solution occurs at a zero critical wave number, through an oscillatory mode. We then show that bifurcating time-periodic solutions correspond to the bounded solutions of a second order complex Ginzburg-Landau equation, where the frequency plays the role of an additional parameter. This result still holds for periodic solutions in a slowly moving frame. We give a reaction diffusion example where coefficients of the above equation are explicitly computed.Dedicated to Klaus Kirchgässner on the occasion of his sixtieth birthday     

Generalization of the Jurdjevic-Quinn theorem and stabilization of bilinear control systems with periodic coefficients: I

The Jurdjevic-Quinn theorem on the global asymptotic stabilization of the origin is generalized to nonlinear time-varying affine control systems with periodic coefficients. The proof is based on the Krasovskii theorem on the global asymptotic stability for periodic systems and the introduced notion of “commutator” for two vector fields one of which is time-varying. The obtained sufficient conditions for stabilization are applied to bilinear control systems with periodic coefficients. We construct a control periodic in t in the form of a quadratic form in x that asymptotically stabilizes the zero solution of a bilinear periodic system with a time-invariant drift.     

Stability of indefinite systems

Absrtract  The paper considers a continuous system

Picardsche Ausnahmewerte bei Lösungen linearer Differentialgleichungen

Every solution w(z)0 of the linear differential equation L_n(w)=w⁽ⁿ⁾+a_n–1(z)w^(n–1)+...+a_o(z)w=0 with polynomial coefficients a_j(z), a_o(z)0, assumes all values a0, infinitely often. Strictly speaking, the only deficient values are zero and infinity. In this paper we study differential equations L_n(w)=0, which have a fundamental system with the following property: Every function of this fundamental system takes the value zero only a finite number of times. It is shown, that such a fundamental system exists if and only if the transformation w(z)=exp(q(z))u(z), where q(z) is a suitable polynomial, transforms the differential equation into one with constant coefficients for u(z).     

On the dynamics of Gowdy space‐times

We study the behavior near the singularity t = 0 of Gowdy metrics. We prove existence of an open dense set of boundary points near which the solution is smoothly “asymptotically velocity term dominated” (AVTD). We show that the set of AVTD solutions satisfying a uniformity condition is open in the set of all solutions. We analyze in detail the asymptotic behavior of “power law” solutions at the (hitherto unchartered) points at which the asymptotic velocity equals zero or one. Several other related results are established. © 2004 Wiley Periodicals, Inc.