RESEARCH OF THE PERIODIC SOLUTION FOR A CLASS OF NONLINEAR DIFFERENTIAL EQUATIONS

ＲＥＳＥＡＲＣＨＯＦＴＨＥＰＥＲＩＯＤＩＣＳＯＬＵＴＩＯＮＦＯＲＡＣＬＡＳＳＯＦ　ＮＯＮＬＩＮＥＡＲＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳＲＥＳＥＡＲＣＨＯＦＴＨＥＰＥＲＩＯＤＩＣＳＯＬＵＴＩＯＮＦＯＲＡＣＬＡＳＳＯＦＮＯＮＬＩＮＥＡＲＤＩＦＦＥ...     

RESEARCH OF THE PERIODIC MOTION AND STABILITY OF TWO-DEGREE-OF-FREEDOM NONLINEAR OSCILLATING SYSTEMS

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｅｓｙｓｔｅｍｏｆｎｏｎｌｉｎｅａｒｏｓｃｉｌｌａｔｉｎｇ ,ｐｅｒｉｏｄｉｃｍｏｔｉｏｎｉｓｏｆｐｒｉｍｅｉｍｐｏｒｔａｎｃｅ .Ｂｕｔｅｘｉｓｔｅｎｃｅｏｆｐｅｒｉｏｄｉｃｓｏｌｕｔｉｏｎｓｉｓａｖｅｒｙｄｉｆｆｉｃｕｌｔｑｕｅｓｔｉｏｎ .Ｌｕｃｋｉｌｙｔｈｅｒｅｅｘｉｓｔｓｏｍｅｋｉｎｄｓｏｆｐｅｒｉｏｄｉｃｓｏｌｕｔｉｏｎｉｎａｃｔｕａｌｐｈｙｓｉｃａｌｓｙｓｔｅｍｓ .Ｔｈｅｒｅｆｏｒｅ ,ｗｅｕｓｕａｌｌｙｃｏｎｃｅｎｔｒａｔｅｄｏｕｒａｔｔｅｎｔｉｏｎｏｎｔ…     

On the <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">s</Emphasis></Superscript> Theory of Hydrostatic Euler Equations

In this paper we study the two-dimensional hydrostatic Euler equations in a periodic channel. We prove the local existence and uniqueness of H ^s solutions under the local Rayleigh condition. This extends Brenier’s (Nonlinearity 12(3):495–512, 1999) existence result by removing an artificial condition and proving uniqueness. In addition, we prove weak–strong uniqueness, mathematical justification of the formal derivation and stability of the hydrostatic Euler equations. These results are based on weighted H ^s a priori estimates, which come from a new type of nonlinear cancellation between velocity and vorticity.     

DISCUSSION ON THE PERIODIC SOLUTIONS FOR HIGHER-ORDER LINEAR EQUATION OF NEU

ＳｉＪｉａｎｇｕｏ（司建国）（ＲｅｃｅｉｖｅｄＭａｙ３０，１９９４，ＣｏｍｍｕｎｉｃａｔｅｄｂｙＬｉｎＺｏｎｇｃｈｉ）ＤＩＳＣＵＳＳＩＯＮＯＮＴＨＥＰＥＲＩＯＤＩＣＳＯＬＵＴＩＯＮＳＦＯＲＨＩＧＨＥＲ－ＯＲＤＥＲＬＩＮＥＡＲＥＱＵＡＴＩＯＮＯＦＮＥＵ...     

Existence of planar flame fronts in convective-diffusive periodic media

We prove the existence of planar travelling wave solutions in a reaction-diffusion-convection equation with combustion nonlinearity and self-adjoint linear part in R ⁿ, n1. The linear part involves diffusion-convection terms and periodic coefficients. These travelling waves have wrinkled flame fronts propagating with constant effective speeds in periodic inhomogeneous media. We use the method of continuation, spectral theory, and the maximum principle. Uniqueness and monotonicity properties of solutions follow from a previous paper. These properties are essential to overcoming the lack of compactness and the degeneracy in the problem.     

On the global asymptotic stability of periodic solutions of a class of ecological system

In this paper, we have made reaearches on the mathematical models which have three populations of mutual action: We have obtained the sufficient conditions respectively for the systems(*)and(**)for existence and uniqueness of single positive periodic solutions which are globally asymptotically stable.     

STABILITY ANALYSIS OF LINEAR AND NONLINEAR PERIODIC CONVECTION IN THERMOHALINE DOUBLE－DIFFUSIVE SYSTEMS

ＳＴＡＢＩＬＩＴＹＡＮＡＬＹＳＩＳＯＦＬＩＮＥＡＲＡＮＤＮＯＮＬＩＮＥＡＲＰＥＲＩＯＤＩＣＣＯＮＶＥＣＴＩＯＮＩＮＴＨＥＲＭＯＨＡＬＩＮＥＤＯＵＢＬＥ－ＤＩＦＦＵＳＩＶＥＳＹＳＴＥＭＳＺｈａｎｇＤｉｍｉｎｇ（张涤明）;ＬｉＬｉｎ（李琳）;ＨｕａｎｇＨ...     

Existence and stability of traveling waves in periodic media governed by a bistable nonlinearity

We prove the existence of multidimensional traveling wave solutions of the bistable reaction-diffusion equation with periodic coefficients under the condition that these coefficients are close to constants. In the case of one space dimension, we prove their asymptotic stability.     

Problem of periodic solutions for neutral functional differential equation

ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＣ(ｋ- 1)2π =ｈ(ｔ) ｜ｈ :Ｒ →Ｒｉｓ (ｋ -1 )＿ｔｈｏｒｄｅｒｃｏｎｔｉｎｕｏｕｓｄｉｆｆｅｒｅｎｔｉａｂｌｅａｎｄｈ(ｔ+ 2π) ≡ｈ(ｔ) ,　　Ｃ2π =ｈ(ｔ) ｜ｈ :Ｒ →Ｒｉｓｃｏｎｔｉｎｕｏｕｓａｎｄｈ(ｔ+ 2π) ≡ｈ(ｔ) ,　　‖ｈ(ｔ)‖ =ｓｕｐｔ∈ [0 ,2π] ｜ｈ(ｔ) ｜ ,　　‖ｈ(ｔ)‖Ｐｋ- 1 =ｍａｘ‖ｈ(ｔ)‖ ,‖ｈ′(ｔ)‖ ,… ,‖ｈ(ｋ- 1) (ｔ)‖ ,　　ｘ(ｍ) (ｔ+ ·) (θ) =ｘ(ｍ) (ｔ+θ)　　θ∈Ｒ (ｍ =0 ,1 ,2 ,… ,ｋ-1 ) .Ｃｌｅａｒｌｙ ,ｘ(ｍ) (ｔ + ·) ∈Ｃ2π, …     

Motion of a Graph by <Emphasis Type="Italic">R</Emphasis>-Curvature

We show the existence of weak solutions to the partial differential equation which describes the motion by R-curvature in R ^d , by the continuum limit of a class of infinite particle systems. We also show that weak solutions of the partial differential equation are viscosity solutions and give the uniqueness result on both weak and viscosity solutions.     

Existence of multidimensional traveling waves in the transport of reactive solutes through periodic porous media

We prove the existence of multidimensional traveling-wave solutions to the scalar equation for the transport of solutes (contaminants) with nonlinear adsorption and spatially periodic convection-diffusion-adsorption coefficients under the assumption that the nonlinear adsorption function satisfies the Lax and Oleinik entropy conditions. In the nondegenerate case, we also prove the uniqueness of the traveling waves. These traveling waves are analogues of viscous shock profiles. They propagate with effective speeds that depend on the periodic porous media only up to their mean states, and are given by an averaged Rankine-Hugoniot relation. This is a direct consequence of the fact that the transport equation is in conservation form. We use the sliding domain method, the continuation method, spectral theory, maximum principles, and a priori estimates. In the degenerate case, the traveling waves are weak solutions of a degenerate parabolic equation and are only Holder continuous. We obtain them by taking suitable limits on the non-degenerate traveling waves. The uniqueness of the degenerate traveling waves is open.     

Periodic Motions of Stokes and Navier–Stokes Flows Around a Rotating Obstacle

We prove the existence and uniqueness of periodic motions to Stokes and Navier–Stokes flows around a rotating obstacle \({D \subset \mathbb{R}^3}\) with the complement \({\Omega = \mathbb{R}^3 \backslash D}\) being an exterior domain. In our strategy, we show the C _b -regularity in time for the mild solutions to linearized equations in the Lorentz space \({L^{3,\infty}(\Omega)}\) (known as weak-L ³ spaces) and prove a Massera-typed Theorem on the existence and uniqueness of periodic mild solutions to the linearized equations in weak-L ³ spaces. We then use the obtained results for such equations and the fixed point argument to prove such results for Navier–Stokes equations around a rotating obstacle. We also show the stability of such periodic solutions.     

Stability of Equilibrium Solutions of Autonomous and Periodic Hamiltonian Systems with <Emphasis Type="Italic">n</Emphasis>-Degrees of Freedom in the Case of Single Resonance

This paper concerns with the study of the stability of an equilibrium solution of an analytic Hamiltonian system in a neighborhood of the equilibrium point with n-degrees of freedom, in the autonomous and periodic case under the presence of a single resonance. Our Main Theorem generalizes several results existing in the literature and we also give a geometrical interpretation of the hypotheses involved there. In particular, our Main Theorem provides necessary and sufficient conditions for the stability of the equilibrium solutions under the existence of a single resonance, depending on the coefficients of the Hamiltonian function.     

Fronts Between Periodic Patterns for Bistable Recursions on Lattices

Bistable space–time discrete systems commonly possess a large variety of stable stationary solutions with periodic profile. In this context, it is natural to ask about the fate of trajectories composed of interfaces between steady configurations with periodic pattern and in particular, to study their propagation as traveling fronts. Here, we investigate such fronts in piecewise affine bistable recursions on the one-dimensional lattice. By introducing a definition inspired by symbolic dynamics, we prove the existence of front solutions and uniqueness of their velocity, upon the existence of their ground patterns. Moreover, the velocity dependence on parameters and the co-existence of several fronts with distinct ground patterns are also described. Finally, robustness of the results to small $C^1$ -perturbations of the piecewise affine map is argued by mean continuation arguments.     

Time Periodic Solutions to the One-Dimensional Nonlinear Wave Equation

This paper is concerned with the time periodic solutions to the one-dimensional nonlinear wave equation with either variable or constant coefficients. By adjusting the basis of L ² function space, we can circumvent the difficulties caused by η _u  = 0 and obtain the existence of a weak periodic solution, which was posed as an open problem by Baubu and Pavel in (Trans Am Math Soc 349:2035–2048, 1997). Finally, an application to the forced Sine-Gordon equation is presented to illustrate the utility of this technique.     

Periodic and “slightly” periodic boundary value problems in elastostatics on bodies bounded in all but one direction

General properties of solutions to elastostatic boundary value problems in which some or all of the functions involved are periodic are studied with particular attention given to problems on bodies unbounded in one direction only. It is shown that, even though the displacement corresponding to a periodic strain may, in a very nontrivial sense, be nonperiodic, it does satisfy a semiperiodicity condition. In addition, a theorem of work and energy is derived for periodic strain states on bodies unbounded in only one direction. This formulation of the theorem of work and energy includes extra terms arising from the possible semiperiodicity of the displacement but only explicitly involves one component of the mean stress. This leads to a discussion of the uniqueness of periodic strain solutions to various boundary value problems. Conditions insuring uniqueness are obtained with the necessity of these conditions demonstrated by counter-examples. The degree to which uniqueness can fail is also studied and is shown to be limited.The next portion of the paper discusses the question of whether periodic boundary value problems must have, in some sense, periodic solutions. This leads naturally to the question of the uniqueness of solutions to boundary value problems which, in themselves, are not necessarily periodic but whose corresponding null boundary value problem is periodic. Positive results to both questions are obtained for several fairly broad classes of problems. Counter-examples are then cited to show the necessity of many of the assumptions used in deriving these results.     

Nonlocalized modulation of periodic reaction diffusion waves: The Whitham equation

In a companion paper, we established nonlinear stability with detailed diffusive rates of decay of spectrally stable periodic traveling-wave solutions of reaction diffusion systems under small perturbations consisting of a nonlocalized modulation plus a localized (L ¹) perturbation. Here, we determine time-asymptotic behavior under such perturbations, showing that solutions consist of a leading order of a modulation whose parameter evolution is governed by an associated Whitham averaged equation.     

On the periodic solutions of differential inclusions and applications

The periodic problem of evolution inclusion is studied and its results are used toestablish existence theorems of periodic solutions of a class of semi-linear differential inclusion.Also existence theorem of the extreme solutions and the strong relaxation theorem are given forthis class of semi-linear differential inclusion. An application to some feedback control systems isdiscussed.     

Second-order nonlinear differential equations with infinite set of periodic solutions

For the differential equation u″ = f(t, u, u′), where the function f: R × R ² → R is periodic in the first variable and f (t, x, 0) ≡ 0, sufficient conditions for the existence of a continuum of nonconstant periodic solutions are found. Published in Neliniini Kolyvannya, Vol. 11, No. 4, pp. 495–500, October–December, 2008.     

Stability and exponential convergence for the Boltzmann equation

We prove existence, uniqueness and stability for solutions of the nonlinear Boltzmann equation in a periodic box in the case when the initial data are sufficiently close to a spatially homogeneous function. The results are given for a range of spaces, including L ¹, and extend previous results in L for the non-homogeneous equation, as well as the more developed L ^p -theory for the spatially homogeneous Boltzmann equation.We also give new L -estimates for the spatially homogeneous equation in the case of Maxwellian interactions.