THE POINTWISE ESTIMATES OF SOLUTIONS FOR A NONLINEAR CONVECTION DIFFUSION REACTION EQUATION

This paper studies the time asymptotic behavior of solutions for a nonlinear convection diffusion reaction equation in one dimension.First,the pointwise estimates of solutions are obtained,furthermore,we obtain the optimal Lp,1≤ p ≤ ＋∞,convergence rate of solutions for small initial data.Then we establish the local existence of solutions,the blow up criterion and the sufficient condition to ensure the nonnegativity of solutions for large initial data.Our approach is based on the detailed analysis of the Green function of the linearized equation and some energy estimates.

POINTWISE ESTIMATES OF SOLUTIONS FOR THE NONLINEAR VISCOUS WAVE EQUATION IN EVEN DIMENSIONS

In this article,we study the pointwise estimates of solutions to the nonlinear viscous wave equation in even dimensions(n≥4).We use the Green’s function method.Our approach is on the basis of the detailed analysis of the Green’s function of the linearized system.We show that the decay rates of the solution for the same problem are different in even dimensions and odd dimensions.It is shown that the solution exhibits a generalized Huygens principle.

THE POINTWISE ESTIMATES TO SOLUTIONS FOR 1-DIMENSIONAL LINEAR THERMO-VISCO-ELASTIC SYSTEM

In this paper, we study the linear thermo-visco-elastic system in one-dimensional space variable. The mathematical model is a hyperbolic-parabolic partial differential system. The solutions of the system show some decay property due to the parabolicity. Based on detailed analysis on the Green function of the system, the pointwise estimates of the solutions are obtained, from which the generalized Huygens’ principle is shown.

The Pointwise Estimates of Solutions to the Cauchy Problem of a Chemotaxis Model

This paper deals with an attraction-repulsion chemotaxis model(ARC) in multi-dimensions. By Duhamel's principle, the implicit expression of the solution to(ARC)is given. With the method of Green's function, the authors obtain the pointwise estimates of solutions to the Cauchy problem(ARC) for small initial data, which yield the W s,p(1 ≤p≤∞) decay properties of solutions.

Global Existence and Pointwise Estimates of Solutions to Generalized Benjamin-Bona-Mahony Equations in Multi Dimensions

This paper is concerned with the global existence and pointwise estimates of solutions to the generalized Benjamin-Bona-Mahony equations in all space dimensions.By using the energy method, Fourier analysis and pseudo-differential operators, the global existence and pointwise convergence rates of the solution are obtained. The decay rate is the same as that of the heat equation and one can see that the solution propagates along the characteristic line.

Large time behavior and pointwise estimates for compressible Euler equations with damping

The Cauchy problem of the compressible Euler equations with damping in multi-dimensions is considered when the initial perturbation in H3-norm is small. First, by using two new energy functionals together with the Green's function and iteration method, we improve the L2-decay rate in Tan and Wang(2013)and Tan and Wu(2012)when(ρ0-ˉρ,m)˙B-s1,∞×˙B-s+11,∞with s∈[0,2]is bounded.In particular,it holds that the density converges to its equilibrium state at the rate(1+t)-34-s2 in L2-norm and the momentum decays at the rate(1+t)-54-s2 in L2-norm.Moreover,under a weaker and more general condition on the initial data,we show that the density and the momentum have different pointwise estimates in dimension d with d 3on both space variable x and time variable t as|Dαx(ρ-ˉρ)|C(1+t)-d2-|α|2(1+|x|21+t)-rwith r>d2and|Dαxm|C(1+t)-d2-|α|+12(1+|x|21+t)-d2 by a more elaborate analysis on the Green’s function.These results improve those in Wang and Yang(2001),where the density and the velocity(the momentum)have the same pointwise estimates.

POINTWISE ESTIMATE OF SOLUTIONS OF ISENTROPIC NMIER-STOKES EQUATIONS IN EVEN SPACE-DIMENSIONS

This paper is concerned with the dissipation of solutions of the isentropic Navier-Stokes equations in even and bigger than two multi-dimensions. Pointwise estimates of the time-asymptotic shape of the solutions are obtained and the generalized Huygan's principle is exhibited. The approch of the paper is based on the detailed analysis of the Green function of Iinearized system. This is used to study the coupling of nonlinear diffesion waves.

ON THE POINTWISE ESTIMATIONS OF APPROXIMATION OF FUNCTIONS AND THEIR DERIVATIVES BY HERMITE INTERPOLATION

The object of this paper is to establish the pointwise estimations of approximation of functions in C^1 and their derivatives by Hermite interpolation polynomials. The given orders have been proved to be exact in gen- eral.

ON POINTWISE ESTIMATE FOR GAMMA OPERATORS

For linear combinations of Gamma operators, if 0<a<2r, 1/2-1/2r≤λ≤1. or 0≤λ≤1/2-1/2r(r≥2),0 <a<r+1/1-λ, we obtain an equivalent theorem with ω(?)(f,t) instead of ω(?)(f,t), where ω(?)(f,t) is the Ditzian-Totik moduli of smoothness.

Weighted approximation of functions with singularities by Bernstein operators

As an important type of polynomial approximation, approximation of functions by Bernstein operators is an important topic in approximation theory and computational theory. This paper gives global and pointwise estimates for weighted approximation of functions with singularities by Bernstein operators. The main results are the Jackson＇s estimates of functions f∈ （Wwλ）2 andre Cw, which extends the result of （Della Vecchia et al., 2004）.

LARGE TIME BEHAVIOR OF SOLUTIONS TO THE PERTURBED HASEGAWA-MIMA EQUATION

The large time behavior of solutions to the two-dimensional perturbed Hasegawa- Mima equation with large initial data is studied in this paper. Based on the time-frequency decomposition and the method of Green function, we not only obtain the optimal decay rate but also establish the pointwise estimate of global classical solutions.

COMONOTONE APPROXIMATION WITH INTERPOLATION AT THE ENDS ON AN INTERVAL

Let a function f E C[-1, 1], changes its monotonisity at the finite collection Y ：= {y1,……, ys} of s points Yi ∈ （-1, 1）. For each n 〉 N（Y）, we construct an algebraic polynomial Pn, of degree 〈 n, which is comonotone with f, that is changes its monotonisity at the same points yi as f, and ｜f（x） - Pn（x）｜ ≤ c（s）ω2 （f1 √1-x^2/n）,x∈ [-1,1] where N（Y） is a constant depending only on Y, c（s） is a constant depending only on s and ω2 （f, t） is the second modulus of smoothness of f.

NONLINEAR EVOLUTION SYSTEMS AND GREEN’S FUNCTION

In this paper, we will introduce how to apply Green＇s function method to get the pointwise estimates for the solutions of Cauchy problem of nonlinear evolution equations with dissipative structure. First of all, we introduce the pointwise estimates of the time-asymptotic shape of the solutions of the isentropic Navier-Stokes equations and show to exhibit the generalized Huygen＇s principle. Then, for other nonlinear dissipative evolution equations, we will only introduce the result and give some brief explanations. Our approach is based on the detailed analysis of the Green＇s function of the linearized system and micro-local analysis, such as frequency decomposition and so on.

Some Remarks on Euler Equations with Damping in Multi-Dimensions

We consider the Cauchy problem of Euler equations with damping. Based on the Green function and energy estimates of solutions, we improve the pointwise estimates and obtainL 1 estimate of solutions.

Pointwise decaying rate of large perturbation around viscous shock for scalar viscous conservation law

In this paper, we consider the large perturbation around the viscous shock of the scalar conservation law with viscosity in one dimension case. We divide the time region into t ≤T0 and t 〉 To for a fixed constant To when applying energy method. Since To is fixed, the case t ≤ To is easy to deal with and when t 〉 To, from the decaying property of the solution, there is a priori estimate for the solution. Thus we can succeed to control the nonlinear term and get the pointwise estimate for the perturbation by the weighted energy method.

POINTWISE ESTIMATE OF BOUNDED SOLUTIONS TO SOME DISCRETE INEQUALITIES INVOLVING INFINITE SUMMATION

Explicit bounds on bounded solutions to a new class of Volterra-type linear and nonlinear discrete inequalities involving infinite sums are established. These inequalities can be viewed as discrete analogues of some Volterra-type inequalities having improper integral functionals,which are new to the literature.

Pointwise Estimate of Solutions to the Three-Dimensional Compressible Magnetohydrodynamic Equations

This paper is devoted to the pointwise estimate of solutions for the initial value problem to the three-dimensional compressible magnetohydrodynamic equations, which models the dynamics of compressible quasi-neutrally ionized fluids under the influence of electromagnetic fields. Based on the detailed analysis of the Green function of the linearized system, we obtain the pointwise estimates of smooth solutions when the initial data is sufficiently small with the algebraic decay to the constant equilibrium. As the by-product, we also show the corresponding pL-estimates of the smooth solutions.

Linearly Compact Difference Scheme for the Two-Dimensional Kuramoto-Tsuzuki Equation with the Neumann Boundary Condition

In this paper,we analyze and test a high-order compact difference scheme numerically for solving a two-dimensional nonlinear Kuramoto-Tsuzuki equation under the Neumann boundary condition.A three-level average technique is utilized,thereby leading to a linearized difference scheme.The main work lies in the pointwise error estimate in H^(2)-norm.The optimal fourth-order convergence order is proved in combination of induction,the energy method and the embedded inequality.Moreover,we establish the stability of the difference scheme with respect to the initial value under very mild condition,however,does not require any step ratio restriction.Extensive numerical examples with/without exact solutions under diverse cases are implemented to validate the theoretical results.