On nonoscillating solutions of Emden-Fowler-type equations

The asymptotic properties of nonoscillating solutions of Emden-Fowler-type equations of arbitrary order are considered. The paper contains the results of the study of the asymptotic properties of solutions with integer-valued asymptotics as well as of solutions arising from the rapid decrease of the coefficient of the equation. To analyze the asymptotic behavior of solutions of the equations, methods of power geometry are used.     

STABILITY OF SOLUTIONS OF FIRST ORDER NONLINEAR NEUTRAL DIFFERENTIAL EQUATIONS WITH OSCILLATORY COEFFICIENTS

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一类奇摄动方程的PLK方法

通过对坐标作包含因变量的非线性泛函的变换,以首项渐近解和相应的坐标变换给出原问题的二阶的近似解,并把这种思想进一步推广到更复杂的非线性方程,用较为简洁的方法求得了一类非线性方程的二阶渐近解.     

Asymptotic Behaviour and Global Existence of Solutions for Some Classes of Nonlinear Parabolic Equations

The paper deals with the asymptotic behaviour and global existence of solutions for some classes of nonlinear parabolic equations in regard to the monotone properties of the nonlinear term. The asymptotic behaviour of the solutions of initial-boundary value problem for nonlinear parabolic equations is studied via the method of differential inequalities in order to obtain oscillation criterion for the solutions. Existence of extremal solutions of semilinear elliptic and parabolic equations is investigated via monotone iterative methods. The extremal solutions are obtained via monotone iterates.     

Large time behaviour of solutions to a class of non-autonomous, degenerate parabolic equations

We consider a class of non-autonomous, degenerate parabolic equations and we study the asymptotic behaviour of the solutions. Even if the equation depends explicitly upon the time, we prove that several asymptotic properties, valid for the autonomous case, are preserved in this more general situation. To our knowledge, it is the first time that the asymptotic behaviour of solutions to non-autonomous equations is studied.     

Asymptotic behavior of solutions of second-order nonlinear impulsive differential equations

In this work, the asymptotic behavior of all solutions of second-order nonlinear ordinary differential equations with impulses is investigated. By impulsive differential inequality and Riccati transformation, sufficient conditions of asymptotic behavior of all solutions of second-order nonlinear ordinary differential equations with impulses are obtained. An example is also inserted to illustrate the impulsive effect.     

Asymptotic Behavior of Solutions of Dynamic Equations

We consider linear dynamic systems on time scales, which contain as special cases linear differential systems, difference systems, or other dynamic systems. We give an asymptotic representation for a fundamental solution matrix that reduces the study of systems in the sense of asymptotic behavior to the study of scalar dynamic equations. In order to understand the asymptotic behavior of solutions of scalar linear dynamic equations on time scales, we also investigate the behavior of solutions of the simplest types of such scalar equations, which are natural generalizations of the usual exponential function.     

Asymptotic forms of positive solutions of third-order Emden-Fowler equations

We obtain asymptotic forms of positive solutions of third-order Emden-Fowler equations. These results improve earlier results for asymptotic behavior of positive solutions considerably.     

Explicit asymptotic limits for a class of discrete Volterra equations

This paper presents explicit asymptotic limits for some solutions of a special kind of Volterra difference equations and exhibits a class of equations having asymptotic equilibrium.     

The dynamics of second-order equations with delayed feedback and a large coefficient of delayed control

The dynamics of second-order equations with nonlinear delayed feedback and a large coefficient of a delayed equation is investigated using asymptotic methods. Based on special methods of quasi-normal forms, a new construction is elaborated for obtaining the main terms of asymptotic expansions of asymptotic residual solutions. It is shown that the dynamical properties of the above equations are determined mostly by the behavior of the solutions of some special families of parabolic boundary value problems. A comparative analysis of the dynamics of equations with the delayed feedback of three types is carried out.     

The dissipation of nonlinear dispersive waves: the case of asymptotically weak nonlinearity

It is proved herein that certain smooth, global solutions of a class of quasi-linear, dissipative wave equations have precisely the same leading order, long-time, asymptotic behavior as the solutions with the same initial data of the corresponding linearized equations. The solutions of the nonlinear equations are shown to be asymptotically self-similar with explicitly determined profiles. The equations considered have homogeneous nonlinearities and homogeneous dispersive and dissipative symbols. By relating these degrees of homogeneous to the leading order asymptotic behavior of the Fourier transform of the initial data near k= 0, different classes of long-time asymptotic behavior are characterized. These results cover the case where dissipation is not asymptotically negligible in comparison with dispersion, and where nonlinear effect are asymptotically negligible in comparison with linear effect, i.e., dissipation and dispersion. They always hold for solutions with "small" initial data. In most circumstances however a new a priori bound on certain negative homogeneous Sobolev norms of solutions is obtained, which implies that any solution, even one which is initially "large" will eventually satisfy the smallness condition, and hence will have the above described asymptotic behavior     

具有两个转向点的大参数奇摄动方程的渐近解

利用匹配法研究了一类具有两个转向点的大参数奇摄动方程,通过Liouville-Green变换和Airy函数分别构造了方程在不同区域的外部解和内层解,得出了方程的渐近解,即解在不同范围内的5个渐近表达式及其5对常数之间的4个匹配条件.     

Asymptotics of the eigenvalues of a boundary value problem for a vector singular differential equation

By analyzing a system of integro-differential equations of the Volterra-Stieltjes type, for large parameter values, we obtain asymptotic formulas for a linearly independent system of solutions of ordinary differential equations with generalized functions in coefficients. These solutions permit one to construct asymptotic formulas for the eigenvalues of a boundary value problem in the case of regular boundary conditions.     

Asymptotically recurrent solutions ofβ-differential equations

In the paper methods from the theory of extensions of dynamical systems are used to studyβ-differential equations whose solutions possess the uniqueness property and depend continuously on the initial data and on the right-hand side of the equation. The Zhikov-Bronshtein theorems concerning asymptotically almost periodic solutions of ordinary differential equations are extended toβ-differential equations (in particular, to total differential equations). Along with asymptotic almost periodicity, we also consider asymptotic recurrence, weak asymptotic distality, and asymptotic distality. To the equations we associate dynamical systems generated by the space of the right-hand sides and the spaces of the solutions and of the initial data of solutions of the equation. Generally, the phase semigroups of the dynamical systems are not locally compact. Translated fromMatermaticheskie Zametki, Vol. 67, No. 6, pp. 837–851, June, 2000.     

Asymptotic Representations for Solutions of One Class of Systems of Quasilinear Differential Equations

We establish asymptotic representations for solutions of one class of systems of differential equations appearing in the investigation of the asymptotic behavior of nth-order quasilinear differential equations.     

Global existence of small solutions to nonlinear evolution equations

We study the global existence and asymptotic behaviour of “small” solutions of a large class of nonlinear partial differential equations. If the nonlinear terms are of high degree the solutions will be asymptotic to solutions of the linear equation.     

差分方程解的稳定性、有界性及概周期解的存在性

作者通过Liapunov泛函建立了一类高维差分方程解一致稳定、一致渐近稳定及指数渐近稳定的充要条件. 此外, 作者还证明了解的一致渐近稳定性蕴含解的有界性, 同时也给出了概周期差分方程存在概周期解的一个充分条件.     

Asymptotic forms of positive solutions of quasilinear ordinary differential equations with singular nonlinearities

In this paper we consider positive solutions of second order quasilinear ordinary differential equations with singular nonlinearities. We obtain asymptotic equivalence theorems for asymptotically superlinear solutions and decaying solutions. By using these theorems, exact asymptotic forms of such solutions are determined. Furthermore, we can establish the uniqueness of decaying solutions as an application of our results.     

Asymptotic Properties of the Electromagnetic Field in the External Schwarzschild Spacetime

. We study the asymptotic behaviour of the solutions to the vacuum Maxwell equations in the external Schwarzschild spacetime. The results are based on the extensive use of geometric considerations and the introduction of generalized energy estimates. We obtain the asymptotic behaviour along the null outgoing directions and we prove also some partial results concerning the behaviour along the timelike curves. Our techniques can be also used to control the asymptotic behaviour of the various derivatives of the Maxwell field and to obtain the asymptotic behaviour of the Weyl tensor fields, solutions of the "spin 2" equations.     

Lame equation and asymptotic higher-order periodic solutions to nonlinear evolution equations

Based on an auxiliary Lame equation and the perturbation method, a direct method is proposed to construct asymptotic higher-order periodic solutions to some nonlinear evolution equations. It is shown that some asymptotic higher-order periodic solutions to some nonlinear evolution equations in terms of Jacobi elliptic functions are explicitly obtained with the aid of symbolic computation.