Derivation of the Linear Elastic String Model from Three-Dimensional Elasticity

We derive a one-dimensional model for the displacement and torsion of an elastic string starting from a cylindrical three-dimensional linearized prestressed elastic body with small diameter. The prestress is due to the prior elastic deformation of an isotropic, homogenous, elastic body. We deduce the scaling of forces by a formal asymptotic expansion. Then we prove that the family of solutions of three-dimensional problems converges to a limit that is the unique solution of the string model. Coefficients of the string model depend on the three-dimensional elasticity coefficients and the tension due to the predeformation.     

On transversal vibrations of an axially moving string with a time-varying velocity

In this paper an initial-boundary value problem for a linear equation describing an axially moving string will be considered for which the bending stiffness will be neglected. The velocity of the string is assumed to be time-varying and to be of the same order of magnitude as the wave speed. A two time-scales perturbation method and the Laplace transform method will be used to construct formal asymptotic approximations of the solutions. It will be shown that the linear axially moving string model already has complicated dynamical behavior and that the truncation method can not be applied to this problem in order to obtain approximations which are valid on long time-scales.     

Perturbation solutions for asymmetric laminar flow in porous channel with expanding and contracting walls

The cases of large Reynolds number and small expansion ratio for the asym- metric laminar flow through a two-dimensional porous expanding channel are considered. The Navier-Stokes equations are reduced to a nonlinear fourth-order ordinary differential equation by introducing a time and space similar transformation. A singular perturbation method is used for the large suction Reynolds case to obtain an asymptotic solution by matching outer and inner solutions. For the case of small expansion ratios, we are able to obtain asymptotic solutions by double parameter expansion in either a small Reynolds number or a small asymmetric parameter. The asymptotic solutions indicate that the Reynolds number and expansion ratio play an important role in the flow behavior. Nu- merical methods are also designed to confirm the correctness of the present asymptotic solutions.     

Convergence of Anisotropically Decaying Solutions of a Supercritical Semilinear Heat Equation

We consider the Cauchy problem for a semilinear heat equation with a supercritical power nonlinearity. It is known that the asymptotic behavior of solutions in time is determined by the decay rate of their initial values in space. In particular, if an initial value decays like a radial steady state, then the corresponding solution converges to that steady state. In this paper we consider solutions whose initial values decay in an anisotropic way. We show that each such solution converges to a steady state which is explicitly determined by an average formula. For a proof, we first consider the linearized equation around a singular steady state, and find a self-similar solution with a specific asymptotic behavior. Then we construct suitable comparison functions by using the self-similar solution, and apply our previous results on global stability and quasi-convergence of solutions.     

Secondary regimes in couette flow

We consider the asymptotic solutions of secondary steady flows in a fluid contained between cylinders rotating in the same direction for large Reynolds numbers.The existence of secondary axisymmetric steady flows in a fluid contained between cylinders rotating in the same direction was shown in [1, 2]. In the following we present the asymptotic behavior of such solutions for the case of large Reynolds numbers. The construction follows the scheme suggested in [3].     

Dynamics of taut strings traveled by train of forces

This paper analyzes the dynamical response of taut strings crossed by systems of traveling forces at constant velocity. Starting from the classic solution for the single moving load, the effect of trains of forces having a step equal to the string length is dealt with. The response is formulated in terms of a linear map, whose reiteration furnishes the discrete-time response, and enables the investigation of the asymptotic behavior of the system. The analytical solution highlights the presence of many critical velocities, for which an instability phenomenon by response accretion may occur. The presence of damping inhibits the onset of instability but also allows to attain large displacements, especially in correspondence of the first critical velocities of the undamped string. Finite-difference numerical solutions confirm the full validity of the proposed analytical solutions. A simple procedure to deduce an improved solution for the problem of the single moving force is outlined in the Appendix.     

Évolution quasi-statique d'un fil souple inextensible sur un plan avec frottement sec : cas discretQuasi-static motion of a supple,homogenous and inextensible string on a horizontal plane with dry friction: discret case

The differential inclusion describing the quasi-static motion of a supple, homogeneous and inextensible string on a horizontal plane with dry friction (Coulomb's law) is a one dimensional evolution model of a continuous medium, with non-linear geometry, obeying a “plastic-rigid” law. With a view to numerical simulation, we treat the discrete case: the string is assimilated to a chain constituted by rigid rods perfectly articulated around ball-joints. We give variational formulation of the problem and prove existence and uniqueness of solutions. We construct an algorithm that describes the instantaneous solutions when the initial configuration of the string is given. Then, some examples are treated. To cite this article: H. Sayah, C. R. Mecanique 332 (2004).     

The boundedness and asymptotic behavior of solutions of differential system of second-order with variable coefficients

In this paper, the differential system of second-order withvariable coefficients is studied. and some criteria of theboundedness and asymptotic behavior for solutions are given.Consider a system of differential equationsdx_1/dt=p_(11)(t)x_1 p_(12)(t)x_2dx_2/dt=p_(21)(t)x_1 p_(22)(t)x_2Now we studg the boundedness and asymptotic behavior of its so-lutions. In the case of Pij(t)being periodic functions. it wasinvestigated by Burdina; in the case of Pij(t) being arbitraryfunctions. it has not been investigated yet. Besides. the me-thod used by Burdina is only oppropriate for the former but notfor the latter case. In this paper we shall give a method whichis appropriate for both cases.     

On the asymptotic behavior of a phase-field model for elastic phase transitions

We study the asymptotic behavior of a one-dimensional, dynamical model of solid-solid elastic transitions in which the phase is determined by an order parameter. The system is composed of two coupled evolution equations, the mechanical equation of elasticity which is hyperbolic and a parabolic equation in the order parameter. Due to the strong coupling and the lack of smoothing in the hyperbolic equation, the asymptotic behavior of solutions is difficult to determine using standard methods of gradient-like systems. However, we show that under suitable assumptions all solutions approach the equilibrium set weakly, while the phase field stabilizes strongly.     

On a Free Boundary Problem for the Curvature Flow with Driving Force

We study a free boundary problem associated with the curvature dependent motion of planar curves in the upper half plane whose two endpoints slide along the horizontal axis with prescribed fixed contact angles. Our first main result concerns the classification of solutions; every solution falls into one of the three categories, namely, area expanding, area bounded and area shrinking types. We then study in detail the asymptotic behavior of solutions in each category. Among other things we show that solutions are asymptotically self-similar both in the area expanding and the area shrinking cases, while solutions converge to either a stationary solution or a traveling wave in the area bounded case. We also prove results on the concavity properties of solutions. One of the main tools of this paper is the intersection number principle, however in order to deal with solutions with free boundaries, we introduce what we call “the extended intersection number principle”, which turns out to be exceedingly useful in handling curves with moving endpoints.     

On the Large Time Behavior of Solutions of Hamilton–Jacobi Equations Associated with Nonlinear Boundary Conditions

In this article, we study the large time behavior of solutions of first-order Hamilton–Jacobi Equations set in a bounded domain with nonlinear Neumann boundary conditions, including the case of dynamical boundary conditions. We establish general convergence results for viscosity solutions of these Cauchy–Neumann problems by using two fairly different methods: the first one relies only on partial differential equations methods, which provides results even when the Hamiltonians are not convex, and the second one is an optimal control/dynamical system approach, named the “weak KAM approach”, which requires the convexity of Hamiltonians and gives formulas for asymptotic solutions based on Aubry–Mather sets.     

Transient spherical flow of non-newtonian power-law fluids in porous media

The transient spherical flow behavior of a slightly compressible non-Newtonian, power-law fluids in porous media is studied. A nonlinear partial differential equation of parabolic type is derived. The diffusivity equation for spherical flow is a special case of the new equation. We obtain analytical, asymptotic and approximate solutions by using the methods of Laplace transform and weighted mass conservation. The structures of asymptotic and approximate solutions are similar, which enriches the theory of one-dimensional flow of non-Newtonian fluids through porous media.     

Persistence,Permanence and Global Stability for an n-Dimensional Nicholson System

For a Nicholson’s blowflies system with patch structure and multiple discrete delays, we analyze several features of the global asymptotic behavior of its solutions. It is shown that if the spectral bound of the community matrix is non-positive, then the population becomes extinct on each patch, whereas the total population uniformly persists if the spectral bound is positive. Explicit uniform lower and upper bounds for the asymptotic behavior of solutions are also given. When the population uniformly persists, the existence of a unique positive equilibrium is established, as well as a sharp criterion for its absolute global asymptotic stability, improving results in the recent literature. While our system is not cooperative, several sharp threshold-type results about its dynamics are proven, even when the community matrix is reducible, a case usually not treated in the literature.     

Asymptotic analysis of boundary-value problems in thin perforated domains with rapidly varying thickness

For a second-order symmetric uniformly elliptic differential operator with rapidly oscillating coefficients, we study the asymptotic behavior of solutions of a mixed inhomogeneous boundary-value problem and a spectral Neumann problem in a thin perforated domain with rapidly varying thickness. We obtain asymptotic estimates for the differences between solutions of the original problems and the corresponding homogenized problems. These results were announced in Dopovidi Akademii Nauk Ukrainy, No. 10, 15–19 (1991). The new results obtained in the present paper are related to the construction of an asymptotic expansion of a solution of a mixed homogeneous boundary-value problem under additional assumptions of symmetry for the coefficients of the operator and for the thin perforated domain.     

On a Rayleigh Wave Equation with Boundary Damping

In this paper an initial-boundary value problem for a weakly nonlinear string (or wave) equation with non-classical boundary conditions is considered. One end of the string is assumed to be fixed and the other end of the string is attached to a dashpot system, where the damping generated by thedashpot is assumed to be small. This problem can be regarded as a simple model describing oscillations of flexible structures such as overhead transmission lines in a windfield. An asymptotic theory for a class ofinitial-boundary value problems for nonlinear wave equations is presented. Itwill be shown that the problems considered are well-posed for all time t. A multiple time-scales perturbation method incombination with the method of characteristics will be used to construct asymptotic approximations of the solution. It will also be shown that all solutions tend to zero for a sufficiently large value of the damping parameter. For smaller values of the damping parameter it will be shown how the string-system eventually will oscillate. Some numerical results are alsopresented in this paper.     

On Boundary Damping for a Weakly Nonlinear Wave Equation

In this paper an initial-boundary value problem for a weakly nonlinear string(or wave) equation with non-classical boundary conditions is considered. Oneend of the string is assumed to be fixed and the other end of the string isattached to a spring-mass-dashpot system, where the damping generated by thedashpot is assumed to be small. This problem can be regarded as a rather simple model describing oscillationsof flexible structures such as suspension bridges or overhead transmission lines in a windfield. A multiple-timescales perturbation method will be usedto construct formal asymptotic approximations of the solution. It will also beshown that all solutions tend to zero for a sufficiently large value of thedamping parameter. For smaller values of the damping parameter it will be shownhow the string-system eventually will oscillate.     

On the dynamics of a string generator: Effect of delay in nonlinear feedback

We investigate the effect of delay in feedback on the oscillation characteristics (amplitude and frequency) of a string generator, which, as is well known, works in a self-induced oscillation mode and is a part of a string accelerometer (a device for measuring the acceleration of ballistic missiles, launch vehicles, and other moving objects). A mathematical model of the dynamics of a string generator is taken in the form of a quasilinear second-order hyperbolic equation with constant delay with respect to one of independent variables (time). For the analysis of the mathematical model, we use the one-frequency asymptotic Krylov-Bogolyubov-Mitropol'skii method (its first and second approximations) of nonlinear mechanics. We show that an increase in the delay in the nonlinear feedback amplifier results in a decrease in the frequency of self-induced oscillations, which transforms the string generator into a low-frequency device. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 2, pp. 168–190, April–June, 2008.     

Front-Like Entire Solutions for Monostable Reaction-Diffusion Systems

This paper is concerned with front-like entire solutions for monostable reaction-diffusion systems with cooperative and non-cooperative nonlinearities. In the cooperative case, the existence and asymptotic behavior of spatially independent solutions (SIS) are first proved. Further, combining a SIS and traveling fronts with different wave speeds and propagation directions, the existence and various qualitative properties of entire solutions are established by using the comparison principle. In the non-cooperative case, we introduce two auxiliary cooperative systems and establish a comparison theorem for the Cauchy problems of the three systems, and then prove the existence of entire solutions via using the comparison theorem, the traveling fronts and SIS of the auxiliary systems. Our results are applied to some biological and epidemiological models. To the best of our knowledge, it is the first work to study the entire solutions of non-cooperative reaction-diffusion systems.     

On a Free Boundary Problem for a Two-Species Weak Competition System

We study a Lotka–Volterra type weak competition model with a free boundary in a one-dimensional habitat. The main objective is to understand the asymptotic behavior of two competing species spreading via a free boundary. We also provide some sufficient conditions for spreading success and spreading failure, respectively. Finally, when spreading successfully, we provide an estimate to show that the spreading speed (if exists) cannot be faster than the minimal speed of traveling wavefront solutions for the competition model on the whole real line without a free boundary.     

Asymptotic Behavior and Orbital Stability of Galactic Dynamics in Relativistic Scalar Gravity

The Nordström–Vlasov system is a relativistic Lorentz invariant generalization of the Vlasov–Poisson system in the gravitational case. The asymptotic behavior of solutions and the non-linear stability of steady states are investigated. It is shown that solutions of the Nordström–Vlasov system with energy greater or equal to the mass satisfy a dispersion estimate in terms of the conformal energy. When the energy is smaller than the mass, we prove the existence and non-linear (orbital) stability of a class of static solutions (isotropic polytropes) against general perturbations. The proof of orbital stability is based on a variational problem associated to the minimization of the energy functional under suitable constraints.