Convergent families of approximate inertial manifolds for nonautonomous evolution equations

I.IntroduCtionAtpresent.autollomousinfillitedimensionaldynamicalsystemshavebeenthoroughlystLldicdilltllcory.andwidelyappliedinpracticel'-'1.Forthe11onautonomouscase,[5--91havesttldicd1ilocxistenceanddimensionestimateofattractorsofnonautonomouscase;[12].hasconsideredtileexislellceofinertialmanifolds.Theoretically,inertialmanifoldisaveryusefulInethodtodiscussthelongtimebehaviorofthesolutionstononautonomousinfinitedimcnsiollaldynamicalsystems.Butitcannotbeexpressedexplicitly.Soitisnotconvenient…     

Asymptotic behaviors of solutions for dissipative quantum Zakharov equations

The dissipative quantum Zakharov equations are mainly studied. The existence and uniqueness of the solutions for the dissipative quantum Zakharov equations are proved by the standard Galerkin approximation method on the basis of a priori estimate. Meanwhile, the asymptotic behavior of solutions and the global attractor which is constructed in the energy space equipped with the weak topology are also investigated.     

The research of longtime dynamic behavior in weakly damped forced KdV equation

I.IntroductionTheKorteweg-deVries(KdV)equationul uu, u.;:=o(l.1)wasinitiallyder1vedasamodelforonedirectionallongwaterwavesofsmal1amplitudepropagatinginachannel.SincetheworkofKorteweganddeVries,ithasbeenshownthatthisequationoccursinalargevarietyofphysicals…     

The hölder continuity of generalized solutions of a class quasilinear parabolic equations

Let A cud B satisfy the Structural conditions (2), the local Hölder continuityinterior to Q=G×(0, T) is proved for the generalized solutions of quasilinearparabolic equations as follows: u₂ - divA(x, t,u,∇u) + B(x, t, u, ∇u)=0     

The asymptotic behavior of solutions to the Kirchhoff equation with a viscous damping term

We study the asymptotic behavior of solutions to an equation describing non-linear vibration of a string with viscosity. In the case when the string is unstretched (the degenerate case), we determine the decay order of solutions by investigating the dynamics near an infinite-dimensional center manifold. Moreover, we classify the asymptotic behavior of all solutions from a dynamical systems point of view. We also deal with the case where the string is stretched (the nondegenerate case).     

LONG TIME BEHAVIOR FOR SOLUTION OF INITIAL BOUNDARY VALUE PROBLEM OF ONE CLASS OF SYSTEMS WITH MULTIDIMENSIONAL INHOMOGENEOUS GBBM EQUATIONS

The following initial-boundary value problem for the systems with multidimensional inhomogeneous generalized Benjamin-Bona-Mahony ( GBBM ) equations is reviewed. The existence of global attractors of this problem was proved by means of a uniform priori estimate for time.     

The fractal structure of attractor for the generalized Kuramoto-Sivashinsky equations

I.IntroductionSincethereexistspectralbarriersandspectralgapconditions,theexistenceofaninertialmanifoldformanynonlineardissipativeevolutionequationsisstillamystery.Recently,Edenetal[5]havediscoveredthatfornonlinearsemigroup,definedbynonlineardissipativeevolutionequationsincludingZDNavier-Stokesequations,thereexistsatinliefractaldimensionalinertialsetwhichmayberepresentedbyaunionoffractillsetsandattractor,ifitisLipschitzcontinuousandissqueezingonacompacti,ositiveinvariantset.Ontileotherhand,S…     

Pullback attractor of 2D nonautonomous g-Navier-Stokes equations with linear dampness

The pullback attractors for the 2D nonautonomous g-Navier-Stokes equations with linear dampness are investigated on some unbounded domains. The existence of the pullback attractors is proved by verifying the existence of pullback D-absorbing sets with cocycle and obtaining the pullback D-asymptotic compactness. Furthermore, the estimation of the fractal dimensions for the 2D g-Navier-Stokes equations is given. Key words pullback attractor, g-Navier-Stokes equation, pullback asymptotic compactness, fractal dimension, linear dampness     

Long time behavior of solutions to coupled Burgers-complex Ginzbury-Landau (Burgers-CGL) equations for flames governed by sequential reaction

This paper studies the existence and long time behavior of the solutions to the coupled Burgers-complex Ginzburg-Landau （Burgers-CGL） equations, which are derived from the nonlinear evolution of the coupled long-scale oscillatory and monotonic instabilities of a uniformly propagating combustion wave governed by a sequential chem- ical reaction, having two flame fronts corresponding to two reaction zones with a finite separation distance between them. This paper firstly shows the existence of the global solutions to these coupled equations via subtle transforms, delicate a priori estimates and a so-called continuity method, then prove the existence of the global attractor and establish the estimates of the upper bounds of Hausdorff and fractal dimensions for the attractor.     

Permanence and asymptotic properties of nonlinear delay difference equations

The asymptotic behavior of a class of nonlinear delay difference equation wax studied . Some sufficient conditions are obtained for permanence and global attractivity . The results can be applied to a claxs of nonlinear delay difference equations and to the delay discrete Logistic model and some known results are included.     

On the asymptotic behavior of solutions of second-order differential equations with nonlinearity close to an exponential

We obtain asymptotic representations for one class of solutions of a second-order differential equation with nonlinearity close to an exponential. Translated from Neliniini Kolyvannya, Vol. 11, No. 4, pp. 541–553, October–December, 2008.     

SERIES PERTURBATIONS APPROXIMATE SOLUTIONS TO N-S EQUATIONS AND MODIFICATION TO ASYMPTOTIC EXPANSION MATCHED METHOD

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Dirichlet problem of higher order elliptic equation with perturbation both in differential operator and in boundary

This paper is the continuation of article [7]. It gives further results about the asymptotic expression for the solution of higher order elliptic equation in the case of boundary perturbation combined with operator perturbation. When unperturbed problemA ₀ is not on the spectrum, the asymptotic expression for the solution of perturbation problemA may be expanded with respect to the small parameter . WhileA ₀ is on the spectrum, the asymptotic expression of the solution contains negative powers of the small parameter . The approximation of arbitrary order to the solution is considered and the recursive formula for the general term and the estimation of remainder term are given.     

Localization and approximation of attractors for the Ginzburg-Landau equation

This paper studies the long-term behavior of solutions to the Ginzburg-Landau partial differential equation. For each positive integerm we explicitly produce a sequence of approximate inertial manifolds _m,j ,j = 1, 2,..., of dimensionm and associate with each manifold a thin neighborhood into which the orbits enter with an exponential speed and in a finite time. Of course this neighborhood contains the universal attractor which embodies the large time dynamics of the equations. The thickness of these neighborhoods decreases with increasingm for a fixed orderj; however, for a fixedm no conclusion can be made about the thickness of the neighborhoods associated to two differentj's. The neighborhoods associated to the manifolds localize the universal attractor and provide computabie large time approximations to solutions of the Ginzburg-Landau equation.     

Compatibility between finite volumes and finite elements using solutions of shallow water equations for substance transport

This paper formulates a finite volume analogue of a finite element schematization of three‐dimensional shallow water equations. The resulting finite volume schematization, when applied to the continuity equation, exactly reproduces the set of matrix equations that is obtained by the application of the corresponding finite element schematization to the continuity equation. The procedure allows the consistent and mass conserving coupling of the finite element Telemac model for three‐dimensional flow with the finite volume Delft3D‐WAQ model for water quality. The work has been carried out as part of a joint development by LNHE and WL∣Delft Hydraulics to explore the mutual interaction of their software. Copyright © 2006 John Wiley & Sons, Ltd.     

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 .     

ON PERIODIC SOLUTIONS OF SEVERAL CLASSES 0F RICCATI'S EQAUTION AND SECOND一ORDER DIFFERENTIAL EQUATIONS

In this paper, the problem on periodic solutions of several classes of Riccati's equation with periodic coefficients is discussed, and the conditions, under which several classes of secondorder equations with periodic coefficients have periodic solutions, are given.     

Regularity and finite dimensionality of attractor for plate equation on R<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

This paper addresses the regularity and finite dimensionality of the global attractor for the plate equation on the unbounded domain. The existence of the attractor in the phase space has been established in an earlier work of the author. It is shown that the attractor is actually a bounded set of the phase space and has finite fractal dimensionality.