THE OPTIMAL LARGE TIME BEHAVIOR OF3D QUASILINEAR HYPERBOLIC EQUATIONS WITH NONLINEAR DAMPING

We are concerned with the large-time behavior of 3D quasilinear hyperbolic equations with nonlinear damping.The main novelty of this paper is two-fold.First,we prove the optimal decay rates of the second and third order spatial derivatives of the solution,which are the same as those of the heat equation,and in particular,are faster than ones of previous related works.Second,for well-chosen initial data,we also show that the lower optimal L^(2) convergence rate of the k(∈[0,3])-order spatial derivatives of the solution is(1+t)^(-(2+2k)/4).Therefore,our decay rates are optimal in this sense.The proofs are based on the Fourier splitting method,low-frequency and high-frequency decomposition,and delicate energy estimates.

THE LARGE TIME BEHAVIOR OF SPECTRAL APPROXIMATION FOR A CLASS OF PSEUDOPARABOLIC VISCOUS DIFFUSION EQUATION

The asymptotic behavior of the solutions to a class of pseudoparabolic viscous diffusion equation with periodic initial condition is studied by using the spectral method. The semidiscrete Fourier approximate solution of the problem is constructed and the error estimation between spectral approximate solution and exact solution on large time is also obtained. The existence of the approximate attractor AN and the upper semicontinuity d（AN,A） → 0 are proved.

Large Time Behavior of a Weak Solution for Non-Newtonian Flow

This paper concerns large time behavior of a regular weak solution for non-Newtonian flow equations. It is shown that the decay of the solution is of exponential type when the force term is equal to zero and the domain is bounded. Moreover, the ratio of the enstrophy over the energy has a limit as time tends to infinity, which is an eigenvaiue of the Stokes operator.

LARGE TIME ASYMPTOTIC BEHAVIOR OF THE COMPRESSIBLE NAVIER-STOKES EQUATIONS IN PARTIAL SPACE-PERIODIC DOMAINS

In this article, we study the large time behavior of the 3-D isentropic compressible Navier-Stokes equation in the partial space-periodic domains, and simultaneously show that the related profile systems can be described by like Navier-Stokes equations with suitable ＂pressure＂ functions in lower dimensions. Our proofs are based on the energy methods together with some delicate analysis on the corresponding linearized problems.

GLOBAL EXISTENCE AND LARGE TIME BEHAVIOR OF SOLUTION TO REACTING FLOW MODEL WITH BOUNDARY EFFECT

The initial boundary value problem (IBVP) for the 3×3 hyperbolic system of reacting flow with source term proposed by R.J.LeVeque and others (see [8]) is considered.It is shown, in the present paper, that if the initial data are a suitable perturbation of a shiftcd shock profile which is suitably away from the boundary, then there exists a unique smooth solution in R2+ to the IBVP of the 3×3 hyperbolic system, which tends to another shifted shock profile of this system as t →∞.

POINTWISE SPACE-TIME BEHAVIOR OF A COMPRESSIBLE NAVIER-STOKES-KORTEWEG SYSTEM IN DIMENSION THREE

The Cauchy problem of compressible Navier-Stokes-Korteweg system in R^(3) is considered here.Due to capillarity effect of material,we obtain the pointwise estimates of the solution in an H^(4)-framework,which is different from the previous results for the compressible Navier-Stokes system in an H^(6)-framework[24,25].Our result mainly relies on two different descriptions of the singularity in the short wave of Green’s function for dealing initial propagation and nonlinear coupling respectively.Our pointwise results demonstrate the generalized Huygens’principle as the compressible Navier-Stokes system.As a corollary,we have an L^(p) estimate of the solution with p>1,which is a generalization for p≥2 in[33].

LARGE TIME BEHAVIOR OF SOLUTIONS TO NONLINEAR VISCOELASTIC MODEL WITH FADING MEMORY

We study the Cauchy problem of a one-dimensional nonlinear viscoelastic model with fading memory. By introducing appropriate new variables we convert the integro-partial differential equations into a hyperbolic system of balance laws. When it is a perturbation of a constant state, the solution is shown time asymptotically approach- ing to predetermined diffusion waves, Pointwise estimates on the convergence details are obtained.

Global Existence and Large Time Asymptotic Behavior of Strong Solution to the Cauchy Problem of 2D Density-Dependent Boussinesq Equations of Korteweg Type

In this paper, we study the Cauchy problem of the density-dependent Boussinesq equations of Korteweg type on the whole space with a vacuum. It is proved that there exists a unique strong solution for the two-dimensional Cauchy problem established that the initial density and the initial temperature decay not extremely slow. Particularly, it is allowed to be arbitrarily large for the initial data and vacuum states for the initial density, even including the compact support. Moreover, when the density depends on the Korteweg term with the viscosity coefficient and capillary coefficient, we obtain a consistent priority estimate by the energy method, and extend the local strong solutions to the global strong solutions. Finally, when the pressure and external force are not affected, we deform the fluid models of Korteweg type, we can obtain the large time decay rates of the gradients of velocity, temperature and pressure.

THE CAUCHY PROBLEM FOR THE CAMASSA-HOLM-NOVIKOV EQUATION

In this paper,we consider the Cauchy problem for the Camassa-Holm-Novikov equation.First,we establish the local well-posedness and the blow-up scenario.Second,infinite propagation speed is obtained as the nontrivial solution u(x,t)does not have compact x-support for any t>0 in its lifespan,although the corresponding u0(x)is compactly supported.Then,the global existence and large time behavior for the support of the momentum density are considered.Finally,we study the persistence property of the solution in weighted Sobolev spaces.

GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR FOR AN 1-D COMPRESSIBLE ENERGY TRANSPORT MODEL

In this article, the global existence and the large time behavior of smooth solutions to the initial boundary value problem for a degenerate compressible energy transport model are established.

Derivation of energy-based base shear force coefficient considering hysteretic behavior and P-delta effects

A modified energy-balance equation accounting for P-delta effects and hysteretic behavior of reinforced concrete members is derived. Reduced hysteretic properties of structural components due to combined stiffness and strength degradation and pinching effects, and hysteretic damping are taken into account in a simple manner by utilizing plastic energy and seismic input energy modification factors. Having a pre-selected yield mechanism, energy balance of structure in inelastic range is considered. P-delta effects are included in derived equation by adding the external work of gravity loads to the work of equivalent inertia forces and equating the total external work to the modified plastic energy. Earthquake energy input to multi degree of freedom（MDOF） system is approximated by using the modal energy-decomposition. Energybased base shear coefficients are verified by means of both pushover analysis and nonlinear time history（NLTH） analysis of several RC frames having different number of stories. NLTH analyses of frames are performed by using the time histories of ten scaled ground motions compatible with elastic design acceleration spectrum and fulfilling duration/amplitude related requirements of Turkish Seismic Design Code. The observed correlation between energy-based base shear force coefficients and the average base shear force coefficients of NLTH analyses provides a reasonable confidence in estimation of nonlinear base shear force capacity of frames by using the derived equation.

DECAY ESTIMATIONS OF THE SOLUTION OF A NONLINEAR PSEUDOPARABOLIC EQUATION

We study the large time behavior of the solution to the initial value problem for a nonlinear pseudoparabolic equation with strong nonlinear terms. The Fourier transform, integral estimate, and the splitting of the frequency domain are used.

ASYMPTOTIC HOMOGENIZATION IN A PARABOLIC SEMILINEAR PROBLEM WITH PERIODIC COEFFICIENTS AND INTEGRABLE INITIAL DATA

In this paper, we study the large time behavior of solutions of the parabolic semilinear equation δtu-div（a（x）△↓u） = -｜u｜^αu in （0,∞） × R^N, where α 〉 0 is constant and a∈ Cb^1（R^N） is a symmetric periodic matrix satisfying some ellipticity assumptions.Considering an integrable initial data u0 and α ∈ （2/N, 3/N）, we prove that the large time behavior of solutions is given by the solution U（t, x） of the homogenized linear problem δtU-div（a^h△↓U）=0,U（0） = C, where a^h is the homogenized matrix of a（x）, C is a positive constant and δ is the Dirac measure at 0.

Nonstandard Numerical Schemes for a Linear Stochastic Oscillator with Additive Noise

In order to simulate a linear stochastic oscillator with additive noise,improved nonstandard optimal(INSOPT) schemes are derived utilizing the nonstandard finite difference(NSFD)technique and the improvement technique.These proposed schemes reproduce long time features of the oscillator solution exactly.Their abilities in preserving the symplecticity,the linear growth property of the second moment and the oscillation property of the solution of the stochastic oscillator system on long time interval are studied.It can be shown that the component { x_n}_(n≥1) of the INSOPT schemes switch signs infinitely many times as n →∞,almost surely.Further,the mean-square convergence order of 1 is obtained for these INSOPT schemes.Finally,numerical experiments illustrate intuitively the results obtained in this paper.

Large Time Behavior of Solutions to a 3D Keller-Segel-Stokes System Involving a Tensor-valued Sensitivity with Saturation

In this paper we deal with the initial-boundary value problem for the coupled Keller-Segel-Stokes system with rotational flux, which is corresponding to the case that the chemical is produced instead of consumed,■subject to the boundary conditions( n-n S(x, n, c) c) · ν = c · ν = 0 and u = 0, and suitably regular initial data(n0(x), c0(x), u0(x)), where ? ? R3is a bounded domain with smooth boundary ??. Here S is a chemotactic sensitivity satisfying |S(x, n, c)| ≤ CS(1 + n)-αwith some CS> 0 and α > 0. The greatest contribution of this paper is to consider the large time behavior of solutions for the system(KSS), which is still open even in the 2D case. We can prove that the corresponding solution of the system(KSS) decays to(■) exponentially, if the coefficient of chemotactic sensitivity is appropriately small. As a precondition to consider the asymptotic behavior, we also show the global existence and boundedness of the corresponding initial-boundary problem KSS with a simplified method. We find a new phenomenon that the suitably small coefficient CSof chemotactic sensitivity could benefit the global existence and boundedness of solutions to the model KSS.

Stability of the Phase Separation State for Compressible Navier-Stokes/Allen-Cahn System

This paper is concerned with the large time behavior of the Cauchy problem for Navier-Stokes/AllenCahn system describing the interface motion of immiscible two-phase flow in 3-D. The existence and uniqueness of global solutions and the stability of the phase separation state are proved under the small initial perturbations. Moreover, the optimal time decay rates are obtained for higher-order spatial derivatives of density,velocity and phase. Our results imply that if the immiscible two-phase flow is initially located near the phase separation state, then under small perturbation conditions, the solution exists globally and decays algebraically to the complete separation state of the two-phase flow, that is, there will be no interface fracture, vacuum, shock wave, mass concentration at any time, and the interface thickness tends to zero as the time t → +∞.

Large time behaviour for the Fokker-Planck equation with general potential

In this paper, we study the large time behaviour of the solution of the Fokker-Planck equation with general potential. For the long range potential, we prove the polynomial decay estimate in time of the solution. For the slowly growing potential, we prove the sub-exponential convergence of the solution toward the equilibrium.

Dangerous Driving Behavior Recognition and Prevention Using an Autoregressive Time-Series Model

Time headway is an important index used in characterizing dangerous driving behaviors. This research focuses on the decreasing tendency of time headway and investigates its association with crash occurrence. An autoregressive（AR） time-series model is improved and adopted to describe the dynamic variations of average daily time headway. Based on the model, a simple approach for dangerous driving behavior recognition is proposed with the aim of significantly decreasing headway. The effectivity of the proposed approach is validated by means of empirical data collected from a medium-sized city in northern China. Finally, a practical early-warning strategy focused on both the remaining life and low headway is proposed to remind drivers to pay attention to their driving behaviors and the possible occurrence of crash-related risks.

Boltzmann Equations with Quantum Effects (1):Long Time Behavior of Spatial Decay Solutions

The Boltzmann equations for Fermi-Dirac particles and Bose-Einstein particles, both in the absence of external force fields, are combined into a more general form called the Boltzmann equation with quantum effects (BQE). It is assumed that the initial data f(x,v,0) satisfies 0≤f(x,v,0)≤cΦ(x,v,0) for a positive constant c and certain types of control functions Φ(x,v,t). Then within a given function space B(Φ), we prove that f(x+tv,v,t) uniformly converges to f ∞(x,v) in a certain norm where f ∞(x,v)= limt→∞f(x+tv,v,t) and different initial data determines different long time limits.

Long Time Behavior of the Fokker-Planck-Boltzmann Equation

This paper is devoted to the existence and long time behavior of the global classical solution to Fokker-Planck-Boltzmann equation with initial data near the absolute Maxwellian.