ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR THE CHAFEE-INFANTE EQUATION

In higher dimension, there are many interesting and challenging problems about the dynamics of non-autonomous Chafee-Infante equation. This article is concerned with the asymptotic behavior of solutions for the non–autonomous Chafee-Infante equation (?u)/(?t)-?u =λ(t)(u-u~3) in higher dimension, where λ(t) ∈ C~1[0, T ] and λ(t) is a positive, periodic function.We denote λ_1 as the first eigenvalue of-?? = λ?, x ∈ ?; ? = 0, x ∈ ??. For any spatial dimension N ≥ 1, we prove that if λ(t) ≤λ_1, then the nontrivial solutions converge to zero,namely, ■ u(x, t) = 0, x ∈ ?; if λ(t) λ_1 as t → +∞, then the positive solutions t→+∞are "attracted" by positive periodic solutions. Specially, if λ(t) is independent of t, then the positive solutions converge to positive solutions of-?U = λ(U-U~3). Furthermore,numerical simulations are presented to verify our results.     

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.     

对流反应扩散方程的稳定化时间间断时空有限元解的误差估计

在流线迎风Petrov-Galerkin（SUPG）稳定化有限元数值格式的基础上,结合时间方向的变分离散,构造对流反应扩散方程的稳定化时间间断时空有限元格式.该类格式在工程上有一些数值模拟应用,但相关文献没有看到类似数值格式的理论证明.本文以Radau点为节点,构造时间方向的Lagrange插值多项式,证明了稳定化有限元解的稳定性,时间最大模、空间L²（Ω）-模误差估计.文中利用插值多项式和有限元方法相结合的技巧,解耦时空变量,去掉了时空网格的限制条件,提供了时间间断稳定化时空有限元方法的理论证明思路,克服了因时空变量统一导致的实际计算时的复杂性.     

一种求解跳系数椭圆标量方程的渐近展开方法的误差分析

本文研究了具有跳系数(或多尺度性质)的椭圆型PDEs问题.利用渐近展开方法,将具有多尺度性质的PDEs模型问题解耦为若干光滑系数(或单尺度性)的PDEs问题进行求解,获得了和经典线性有限元方法在.0意义下同样的误差阶,表明了该方法的有效性.     

ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR A CLASS OF DELAY DIFFERENCE EQUATION

We propose a class of delay difference equation with piecewise constant nonlinearity. Such a delay difference equation can be regarded as the discrete analog of a differential equation. The convergence of solutions and the existence of asymptotically stable periodic solutions are investigated for such a class of difference equation.     

A POSTERIORI ERROR ESTIMATES FOR FINITE ELEMENT APPROXIMATIONS OF THE CAHN-HILLIARD EQUATION AND THE HELE-SHAW FLOW

This paper develops a posteriori error estimates of residual type for conforming and mixed finite element approximations of the fourth order Cahn-Hilliard equation ut ＋ △（ε△Au-ε^-1f（u）） = 0. It is shown that the a posteriori error bounds depends on ε^-1 only in some low polynomial order, instead of exponential order. Using these a posteriori error estimates, we construct at2 adaptive algorithm for computing the solution of the Cahn- Hilliard equation and its sharp interface limit, the Hele-Shaw flow. Numerical experiments are presented to show the robustness and effectiveness of the new error estimators and the proposed adaptive algorithm.     

ASYMPTOTIC SOLUTIONS OF THE ELASTIC EQUATION IN THE CASE OF TOTAL REFLECTION

In this paper we construct outgoing asymptotic solutions of the anisotropic elastic wave equation coinciding with given data on the boundary when total reflection happens. In this case the solutions consist of the usual hyperbolic parts and the elliptic parts exponentially decreasing in the normal direction to the boundary. The main assertion is that sum of those parts can cover any boundary data because of the elasticity. The proof is reduced to the complex analysis of the symbol of the elastic operator. By means of the asymptotic solutions we can make parametrices and study propagation of the singularities in the case of total reflection.

     

ERROR ESTIMATES OF GALERKIN METHOD FOR HIGH DIMENSIONAL STEADY STATE KURAMOTO-SIVASHINSKY EQUATION

1　ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｅｐａｐｅｒ,ｗｅｃｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇＫ -Ｓｅｑｕａｔｉｏｎ (ｆｏｒｅｘａｍｐｌｅ,ｓｅｅ [1 ] -[3]ａｎｄｒｅｆｅｒｅｎｃｅｓｔｈｅｒｅｉｎ)ｉｎｓｐａｃｅｄｉｍｅｎｓｉｏｎ≥ 3:Δ2 ｕ+Δｕ+ 12 ｜ ｕ｜2 -12 Ｎｉ=1ｌｉ∫ＩＮ｜ ｕ｜2 ｄｘ=0 ,ｘ∈ＩＮ,(1 )ｗｉｔｈＬ ｐｅｒｉｏｄｉｃｂｏｕｎｄａｒｙｃｏｎｄｉｔｉｏｎｓ,Ｌ =(ｌ1,ｌ2 ,… ,ｌＮ)ｂｅｉｎｇｔｈｅｓｉｚｅｏｆａｔｙｐｉｃａｌｐａｔｔｅｒｎｃｅｌｌ (ａｌｓｏｂｉｆｕｒｃａｔｉｏｎｐａｒ…     

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 → ∞.     

EXISTENCE AND ASYMPTOTIC BEHAVIOR OF THE SOLUTIONS FOR A NONLINEAR ELLIPTIC EQUATION ARISING IN ASTROPHYSICS

The author first analyzes the existence of ground state solutions and cylindrically symmetric solutions and then the asymptotic behavior of the ground state solution of the equation -△u=φ(r)up-1,u>0 in RN, u ∈ D1,2(RN),where N≥ 3,x = (x',z)∈ RK×RN-K,2≤K≤N,r =|x'|.It is proved that for 2(N -s)/(N-2) < p < 2* = 2N/(N -2),0 < s < 2, the above equation has a ground state solution and a cylindrically symmetric solution. For p=2*, the above equation does not have a ground state solution but a cylindrically symmetric-solution, and when p close to 2*, the ground state solutions are not cylindrically symmetric. On the other hand, it is proved that as p close to 2*, the ground state solution up has a unique maximum point xp = (x'p,zp) and as p→2*, |x'p|→r0 which attains the maximum of φ on RN.The asymptotic behavior of ground state solution up is also given, which also deduces that the ground state solution is not cylindrically symmetric as p goes to 2*.     

ASYMPTOTIC ERROR EXPANSIONS OF QUADRATIC SPLINE COLLOCATION SOLUTIONS FOR TWO-POINT BOUNDARY VALUE PROBLEMS

In this paper, we consider the following problem:The quadratic spline collocation, with uniform mesh and the mid-knot points are taken as the collocation points for this problem is considered. With some assumptions, we have proved that the solution of the quadratic spline collocation for the nonlinear problem can be written as a series expansions in integer powers of the mesh-size parameter. This gives us a construction method for using Richardson's extrapolation. When we have a set of approximate solution with different mesh-size parameter a solution with high accuracy can he obtained by Richardson's extrapolation.     

ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF NONLINEAR NEUTRAL DIFFERENTIAL EQUATION

The boundedness of the every solution and the asymptotic behavior of all solutions of the nonlinear neutral delay differential equation [x(t) - P(t)x(t - t)]' Q1 (t)f(x{t-σ1))-Q2(t)f(x(t -σ2))=0,t≥t0 are investigated, whereτ,σ1,σ2∈(0,∞), P∈C([t0,∞),R), and Q1,Q2∈C([t0,∞),R), f∈C(R,R). The sufficient conditions obtained improve the existing results in the literatures.     

ERROR ESTIMATES FOR THE RECURSIVE LINEARIZATION OF INVERSE MEDIUM PROBLEMS

This paper is devoted to the mathematical analysis of a general recursive linearization algorithm for solving inverse medium problems with multi-frequency measurements. Under some reasonable assumptions, it is shown that the algorithm is convergent with error estimates. The work is motivated by our effort to analyze recent significant numerical results for solving inverse medium problems. Based on the uncertainty principle, the recursive linearization allows the nonlinear inverse problems to be reduced to a set of linear problems and be solved recursively in a proper order according to the measurements. As an application, the convergence of the recursive linearization algorithm [Chen, Inverse Problems 13（1997）, pp.253-282] is established for solving the acoustic inverse scattering problem.     

Dynamo抛物方程在外区域对初始时刻几何的误差估计

本文导出了从Dynamo理论中提出的抛物方程的解在外区域上对初始时刻几何的连续依赖性.     

Dynamo抛物方程在外区域对初始时刻几何的误差估计

本文导出了从Dynamo理论中提出的抛物方程的解在外区域上对初始时刻几何的连续依赖性.     

ASYMPTOTIC BEHAVIOR AND EXISTENCE OF POSITIVE SOLUTIONS FOR A NEUTRAL DIFFERENCE EQUATION

１ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒｔｈｅｎｅｕｔｒａｌｄｉｆｅｒｅｎｃｅｅｑｕａｔｉｏｎ△（ｘｎ－ｃｘｎ－ｍ）＋ｐｎｘｎ－ｋ＝０，ｎ＝Ｎ，Ｎ＋１，Ｎ＋２，…，（１）ｗｈｅｒｅｃａｎｄｐｎａｒｅｒｅａｌｎｕｍｂｅｒｓ，ｋ，ｍａｒｅｐｏｓｉｔｉｖ...     

THE ERROR ESTIMATES OF HALLEY''''S METHOD

In this paper we give an almost sharp error estimate of Halley's iteration for the majorizing sequence. Compared with the corresponding results in [6,14], it is far better. Meanwhile,the convergence theorem is established .for Halley's iteration in Banach spaces.