ASYMPTOTIC BEHAVIOUR OF EIGENVALUES FOR THE DISCONTINUOUS BOUNDARY-VALUE PROBLEM WITH FUNCTIONAL-TRANSMISSION CONDITIONS

In this study, the boundary-value problem with eigenvalue parameter generated by the differential equation with discontinuous coefficients and boundary conditions which contains not only endpoints of the considered interval, but also point of discontinuity and linear functionals is investigated. So, the problem is not pure boundary-value. The authors single out a class of linear functionals and find simple algebraic conditions on coefficients, which garantee the existence of infinit number eigenvalues. Also the asymptotic formulas for eigenvalues are found.

THE EXISTENCE AND GLOBAL OPTIMAL ASYMPTOTIC BEHAVIOUR OF LARGE SOLUTIONS FOR A SEMILINEAR ELLIPTIC PROBLEM

By Karamata regular variation theory and constructing comparison functions, the author shows the existence and global optimal asymptotic behaviour of solutions for a semilinear elliptic problem Δu = k（x）g（u）, u 〉 0, x ∈ Ω, u｜δΩ =＋∞, where Ω is a bounded domain with smooth boundary in R^N; g ∈ C^1[0, ∞）, g（0） = g＇（0） = 0, and there exists p 〉 1, such that lim g（sξ）/g（s）=ξ^p, ↓Aξ 〉 0, and k ∈ Cloc^α（Ω） is non-negative non-trivial in D which may be singular on the boundary.

ASYMPTOTIC BEHAVIOUR AND EXPONENTIAL STABILITY FOR THERMOELASTIC PROBLEM WITH LOCALIZED DAMPING

A semi-linear thermoelastic problem with localized damping is considered, which is one of the most important mathematical models in material science. The existence and decays exponentially to zero of solution of this problem are obtained. Moreover, the existence of absorbing sets is achieved in the non-homogeneous case. The result indicates that the system which we studied here is asymptotic stability.

Asymptotic behaviour of solution for fourth order wave equation with dispersive and dissipative terms

This paper studies the initial boundary value problem of fourth order wave equation with dispersive and dissipative terms. By using multiplier method, it is proven that the global strong solution of the problem decays to zero exponentially as the time approaches infinite, under a very simple and mild assumption regarding the nonlinear term.

ON THE ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF CERTAIN FIFTH-ORDER ORDINARY DIFFERENTIAL EQUATIONS

The sufficient conditions are given for all solutions of certain non- autonomous differential equation to be uniformly bounded and convergence to zero as t →∞ ?. The result given includes and improves that result obtained by Abou-El-Ela & Sadek .

Oscillatory and Asymptotic Behaviour of Fourth Order Nonlinear Difference Equations

This paper considers a class of fourth order nonlinear difference equations Δ2（rnΔ2yn）+ f（n,yn）=0,n∈N（n0）,where f（n,y）may be classified as superlinear,sublinear,strongly super- linear and strongly sublinear.In superlinear and sublinear cases,necessary and sufficient conditions are obtained for the difference equation to admit the existence of nonoscillatory solutions with special asymptotic properties.In strongly superlinear and strongly sublinear cases,sufficient conditions are given for all solutions to be oscillatory.

ASYMPTOTIC BEHAVIOUR OF THE SOLUTIONS OF A CERTAIN FOURTH-ORDER DIFFERENTIAL EQUATION

The present paper establishes sufficient conditions for the uniformly bounded of any solution of (1.1) and which tend to zero as t → ∞.

Asymptotic Behaviour of Solutions to the Navier-stokes Equations of a Two-dimensional Compressible Flow

In this paper, we are concerned with the asymptotic behaviour of a weak solution to the Navier-Stokes equations for compressible barotropic flow in two space dimensions with the pressure function satisfying p（g） = aglog^d（g） for large g. Here d 〉 2, a 〉 0. We introduce useful tools from the theory of Orlicz spaces and construct a suitable function which approximates the density for time going to infinity. Using properties of this function, we can prove the strong convergence of the density to its limit state. The behaviour of the velocity field and kinetic energy is also briefly discussed.

ASYMPTOTIC BEHAVIOUR OF SUPERSONIC FLOW PAST ACONVEX COMBINED WEDGE

The supersonic flow past a convex combined wedge is discussed. Here the surface of the wedge is composed of two straight lines connected by a convex smooth curve. Under the assumptions that the shock is weak, the vertex of the wedge is less than a critical value and the difference of the slope of these two lines is small, the author proves the global existence of solution with shock front and obtains the asymptotic bebaviour of the solution.

ON THE ASYMPTOTICAL BEHAVIOUR OF SOLUTIONS OF A CLASS OF NONLINEAR CONTROL SYSTEMS

In this paper, the asymptotical behaviour solutions of a class of nonlinear control systems are studied. By establishing infinite integrals along solutions of the system and drawing support from a LaSalle's invariance principle of integral form, criteria of dichotomy and global asymptotical behaviour of solutions are obtained. This work is an improvement and further extension of research methods and results of A. S. Aisagaliev.

ASYMPTOTIC BEHAVIOUR OF A DIFFUSION SYSTEM WITH TIME DELAY IN POPULATION DYNAMICS

In this paper we consider the initial boundary value problem for a class of reaction-diffusion system with time delay in population dynamics. We prove the existence and uniqueness of bounded nonnegative solution for the initial boundary value problem and discuss the asymptotic behaviour of the solution.

SOME RECENT PROGRESS ON STOCHASTIC HEAT EQUATIONS

This article attempts to give a short survey of recent progress on a class of elementary stochastic partial differential equations (for example, stochastic heat equations) driven by Gaussian noise of various covariance structures. The focus is on the existence and uniqueness of the classical (square integrable) solution (mild solution, weak solution). It is also concerned with the Feynman-Kac formula for the solution;Feynman-Kac formula for the moments of the solution;and their applications to the asymptotic moment bounds of the solution. It also briefly touches the exact asymptotics of the moments of the solution.

ON A SECOND ORDER DISSIPATIVE ODE IN HILBERT SPACE WITH AN INTEGRABLE SOURCE TERM

Asymptotic behaviour of solutions is studied for some second order equations including the model casex（t） ＋γx（t） ＋ ↓△φb（x（t）） = h（t） with γ 〉 0 and h ∈ L1（O, ＋∞; H）, φ being continuouly differentiable with locally Lipschitz continuous gradient and bounded from below. In particular when φ is convex, all solutions tend to minimize the potential φ as time tends to infinity and the existence of one bounded trajectory implies the weak convergence of all solutions to equilibrium points.

A NOTE ON QUASI-P RADICAL OF ASSOCIATIVE RINGS

In this paper, the relationship between the time dependent solutions and steady state solutions of the semiconductor equations affected by magnetic field is considered. A decay estimate is proved by a series of estimates on the solutions under some condition.

Decay Rate of Solutions to Hyperbolic System of First Order

In this paper we generalize the global Sobolev inequality introduced by Klainerman in studying wave equation to the hyperbolic system case.We obtain several decay estimates of solutions of a hyperbolic system of first order by different norms of initial data.In particular,the result mentioned in Theorem 1.5 offers an optimal decay rate of solutions,if the initial data belongs to the assigned weighted Sobolev space.In the proof of the theorem we reduce the estimate of solutions of a hyperbolic system to the corresponding case for a scalar pseudodifferential equation of the first order,and then establish the required estimate by using microlocal analysis.

Oscillation of Positive Steady State for Lotka-Volterra Time-Delay Control System with Diffusion

The oscillation of positive steady state and asymptotic behaviour of nonoscillatory solution for Lotka-Volterra time-delay control system with diffusion is discussed in the paper. The sufficient conditions for oscillation of positive steady state are given.