AN ASYMPTOTIC SOLUTION OF THE NONLINEAR REDUCED WAVE EQUATION

1 IntroductionIn the theory of nonlinear optics[1], one is led to consider a nonlinear reduced wave equationwhere k is the wave number and n = n'(IuI') is a function of intensity Of the field and is calledthe index of refraction.We consider the case when the optical beam propagates in a quadratic index nledia sothat n'(luI') can be written aswhere no and n1 are constants.With IuI = O(k--'), d > 0, and following the discussion of this equation in two dimensionalspace (see [2]), we writeSubst…     

热传导方程解在边界附近的渐近行为

考虑热传导方程的初边值问题的解。当初值与边值“不相容”时。由于热传导方程的特性这个解可以在很短时间内变得光滑。并形成一个边界层，本文将通过上、下解的控制给出解在边界附近变化的渐进行为．     

THE ASYMPTOTIC BEHAVIOR OF SOLUTION FOR A CLASS OF STRONGLY NONLINEAR NON-AUTONOMOUS EQUATION

In this paper, using the singularly perturbed theory and the boundary layer corrective method, the asymptotic behavior of solution for a class of strongly nonlinear non-autonomous equations and the infection for asymptotic behavior of the solution with regard to the boundary condition are studied. According to the different regions of the boundary value, the asymptotic expansions of the solution for the original problem are obtained simply and conveniently.     

一类二阶非线性微分方程解的振动性与渐近性

本文研究了二阶非线性微分方程的解的振动性与渐近性.其中σ是一个偶数与奇数的正商.所得的结果是新的,其中之一修正了Wong的结果.     

一类非线性发展方程弱解的长时间性态

本文分别在全空间和半空间两种情形下讨论一类非线性发展方程弱解的长时间性态．证明了其弱解在L^2范数下衰减速率和线性热方程一致，分别为t^-n/2(1-r-1/2)和t^-n/2(1/r-1/2)-1/2。     

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*.     

一类非线性积分微分方程解及其构造

本文讨论了核反应堆动力学中一类具缓发中子的非线性积分微分方程．在一定的条件下证明了该方程在p空间中Mild解、强解、局部解的存在唯一性，以及用迭代法求解的合理性，并估计了收敛速度．     

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO THE SYSTEM OF ONE-DIMENSIONAL NONLINEAR THERMOVISCOELASTICITY

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LONG TIME ASYMPTOTIC BEHAVIOR OF SOLUTION OF DIFFERENCE SCHEME FOR A SEMILINEAR PARABOLIC EQUATION (I)

1.IntroductionLetfi=(0,l),f(x)eL'(fl),"o(x)EH'(fl)nHI(fl),op(u)=u',weconsiderthefollowinginitial-boundaryvalueproblem:Bytheusualapproach[1--4]wecangettheglobalexistenceofthesolutionof(1.1),furthermore,thesolutionof(l.l)convergestothesolutionofthefollowingstationaryproblem(1.2)ast-co.l)Th.ProjectissupportedbyChinaPostdoctoralFoundationandScienceFOundationofChinaAcademyofEngineeringPhysics.Inthispaperweprovethatthesolutionofimplicitdifferenceschemefor(1.1)convergestothesolutionofdifferenc…     

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.     

ASYMPTOTIC BEHAVIOR OF POSITIVE RADIAL SOLUTIONS FOR THE GENERALIZED EMDEN-FOWLER EQUATION WITH APPLICATION

In this paper, we consider a class of semilinear parabolic differential equation where nonlinear term has local bounded coefficients. Under some assumptions, we get the existence and uniqueness of solution. Terminate to the time variable, we obtain the so called generalized Emden-Fowler equation and the asymptotic behavior of positive radial solutions have been given in all dimensions. At the end of this paper, we give its application to critical branching Brownian motion (also called measure-valued branching processes).     

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

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LONG TIME ASYMPTOTIC BEHAVIOR OF SOLUTION OF IMPLICIT DIFFERENCE SCHEME FOR A SEMI-LINEAR PARABOLIC EQUATION

In this paper, the solution of back-Euler implicit difference scheme for a semi-linea rparabolic equation is proved to converge to the solution of difference scheme for the corresponding semi-linear elliptic equation as t tends to infinity. The long asymptotic behavior of its discrete solution is obtained which is analogous to that of its continuous solution. At last, a few results are also presented for Crank-Nicolson scheme.     

带Neumann边界条件的相场方程解的渐近性态

利用Simon-Lojasiewicz型不等式,获得了带Neumann边界条件的相场方程解的渐近性态.本文所考虑的稳态问题含有非局部项.     

ASYMPTOTIC BEHAVIOR OF SINGULARLY PERTURBED SOLUTION FOR A CLASS OF ROBIN PROBLEM OF NONLINEAR NON-AUTONOMOUS EQUATION

In this paper, using the layer theory, a class of singularly perturbed Robin problem of nonlinear non-autonomous equation is considered. According to the different regions of the boundary value, the asymptotic behaviors of solution for the original problem are obtained simply and conveniently.     

非线性抛物方程耦合的离散化及其误差分析

0　引　　言对于线性抛物型初边值问题的有限元与边界元耦合法,我们已作过研究(可参见[2],[3]).然而,许多实际问题,如流体、对流扩散、热辐射及热传输等涉及到非线性问题,因此研究非线性问题的数值方法显得尤为重要.近年来,G.N.Gatica与G.C.Hsiao已将有限元(FEM)与边界元(BEM)耦合法拓广到非线性椭圆问题(如[5],[6]),但如何应用FEM与BEM耦合法来处理非线性抛物型初边值问题,就作者所知迄今为止尚属空白.这里我们试图对此进行研究.设Ω为R2中一有界单连通区域,其边界为Γ:=Ω.Ωc:=R2\Ω为闭区域Ω的补区域,I:=(0,T].我们考虑如…     

一类非线性差分方程正解的不存在性

本文研究一类非线性差分方程及其对偶方程,给出了方程不存在正解的充分条件,所得结论改进了[1]的结果。     

一类非线性无界时滞差分方程正解的不存在性

研究一类非线性无界时滞差分方程及其对偶方程，给出了方程不存在正解的充分条件，所得结论改进了有关的结果。     

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO THE HEAT EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS

The purpose of this paper is to investigate the stability and asymptotic behav-ior of the time-dependent solutions to a linear parabolic equation with nonlinear boundarycondition in relation to their corresponding steady state solutions. Then, the above resultsare extended to a semilinear parabolic equation with nonlinear boundary condition by an-alyzing the corresponding eigenvalue problem and using the method of upper and lowersolutions.     

EXISTENCE AND UNIQUENESS OF THE ENTROPY SOLUTION TO A NONLINEAR HYPERBOLIC EQUATION

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