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 BEHAVIOR OF SOLUTIONS OF QUANTUM HYDRODYNAMIC MODEL FOR SEMICONDUCTORS

A one-dimensional quantum hydrodynamic model （or quantum Euler-Poisson system） for semiconductors with initial boundary conditions is considered for general pressure-density function. The existence and uniqueness of the classical solution of the corresponding steady-state quantum hydrodynamic equations is proved. Furthermore, the global existence of classical solution, when the initial datum is a perturbation of t he steadystate solution, is obtained. This solution tends to the corresponding steady-state solution exponentially fast as the time tends to infinity.

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 →∞.

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.

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].

LONG-TIME BEHAVIOR OF TRANSIENT SOLUTIONS FOR CELLULAR NEURAL NETWORK SYSTEMS

By establishing concept an transient solutions of general nonlinear systems converging to its equilibrium set, long-time behavior of solutions for cellular neural network systems is studied. A stability condition in generalized sense is obtained. This result reported has an important guide to concrete neural network designs.

LARGE TIME BEHAVIOR OF SOLUTION TO NONLINEAR DIRAC EQUATION IN 1+1 DIMENSIONS

This paper studies the large time behavior of solution for a class of nonlinear massless Dirac equations in R^(1+1). It is shown that the solution will tend to travelling wave solution when time tends to infinity.

Small-volume solution current-time behavior study for application in reverse iontophoresis-based non-invasive blood glucose monitoring

The reverse iontophoresis-based glucose monitoring circumstance is similar to the small-volume solution in which mass diffusion controls the current response of the electrochemical biosensors.In this study,the law of mass transfer in this type of solution was analyzed and a mathematic model was established to depict the current-time behavior of the fabricated planar electrode used in the non-invasive meter designed by ourselves.A small-volume glucose solution was directly constructed on the electrode to simulate the reverse iontophoresis-based sensing condition.The correctness of the model was demonstrated by chronoamperometry.Animal assay was subsequently carried out to verify the practicality of the model in determination of blood glucose.The results processed by the new method accurately traced the authentic value,confirming the advantage of the new method and the potential in clinical analysis.

Large time behavior of Euler-Poisson system for semiconductor

In this note,we present a framework for the large time behavior of general uniformly bounded weak entropy solutions to the Cauchy problem of Euler-Poisson system of semiconductor devices.It is shown that the solutions converges to the stationary solutions exponentially in time.No smallness and regularity conditions are assumed.

Large-time Behavior of Solutions to the Stokes Approximation Equations for Two Dimensional Compressible Flows

In this paper, we study the large time asymptotic behavior of solutions to both the Cauchy problem and the exterior problem of the Stokes approximation equations of two dimensional compressible flows.

GLOBAL EXISTENCE AND LONG-TIME BEHAVIOR FOR THE STRONG SOLUTIONS IN H2 TO THE 3D COMPRESSIBLE NEMATIC LIQUID CRYSTAL FLOWS

In this paper, we investigate the global existence and long time behavior of strong solutions for compressible nematic liquid crystal flows in threedimensional whole space. The global existence of strong solutions is obtained by the standard energy method under the condition that the initial data are close to the constant equilibrium state in H2-framework. If the initial datas in Ll-norm are finite additionally, the optimal time decay rates of strong solutions are established. With the help of Fourier splitting method, one also establishes optimal time decay rates for the higher order spatial derivatives of director.

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.

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.

Large Time Behavior of Entropy Solutions to Some Hyperbolic System with Dissipative Structure

In this paper the large time behavior of the global L∞ entropy solutions to the hyperbolic system with dissipative structure is investigated. It is proved that as t → ∞ the entropy solutions tend to a constant equilibrium state in L2 norm with exponential decay even when the initial values are arbitrarily large. As an illustration, a class of 2 × 2 system is studied.

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.

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.

Large Time Behavior of Solutions to the Generalized Hall-Magnetohydrodynamics System

In this paper,we investigate the large time behavior of solutions to the three dimensional generalized Hall-magnetohydrodynamics(Hall-MHD)system in the spacesχ^(s)(R^(3)).We obtain that the temporal decay rate is‖(u,b)(t)‖χ^(1−2α)+‖(u,b)(t)‖χ^(1−2β)+‖(u,b)(t)‖χ^(2−2α)+‖(u,b)(t)‖χ^(2−2β)≤(1 t)^(-(5-4max{α,β}/4max{α,β})with 1/2≤α,β≤1 for the small global solution by using Fourier splitting method.The parametersαandβare the fractional dissipations corresponding to the velocity and magnetic field,respectively.

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.