SOME BOUNDARY PROBLEMS OF THE SPIN SYSTEMS AND THE SYSTEMS OF FERRO-MAGNETIC CHAIN Ⅰ. NONLINEAR BOUNDARY PROBLEMS

§1. Introduction The nonlinear system of partial differential equations and the corresponding system with diffusion term are interesting and meaningful both in physical and mathematical point of view, where z= (u, v, w) and f(x, t, z) are two three-dimensional vector functions and"× "denotes the cross product operator of two three-dimensional vectors. The system (1) contains the so-called Landau- Lifschitz equation of isotropic Heigenberg ferro-magnetio chain as the     

BOUNDARY VALUE PROBLEMS FOR THE EQUILIBRIUM SYSTEMS OF FERRO-MAGNETIC CHAIN

ＢＯＵＮＤＡＲＹＶＡＬＵＥＰＲＯＢＬＥＭＳＦＯＲＴＨＥＥＱＵＩＬＩＢＲＩＵＭＳＹＳＴＥＭＳＯＦＦＥＲＲＯ－ＭＡＧＮＥＴＩＣＣＨＡＩＮＳｈｅｎＹａｏｔｉａｎ（沈尧天）ＹａｎＳｈｕｓｅｎ（严树森）（Ｄｅｐｔ：ｏｆＡｐｐｌ．Ｍａｔｈ，ＳｏｕｔｈＣｈｉｎａＵ...     

THE BOUNDARY VALUE PROBLEMS FOR THE HIGHER ORDER QUASILINEAR PARABOLIC AND ELLIPTIC SYSTEMS SATISFYING GENERAL BOUNDARY CONDITIONS

In this paper, the existence and uniqueness of the boundary value problems for the higher order quasilinear parabolic systems satisfying general boundary conditions and initial value condition are considered. We have concluded the problems to the following: if all the solutions of a family of problems of the same type which are derived from a substitution of τf(s, t, u, ...,Dx~(2b-1)u), τgj(y, t, u) and τ(x) (0≤τ≤1) for f, g, and φ respectively are uniformly bounded, then the original problems have a unique solution in H~(2b a,1 a/2b)((?)_T). Under assumption that the linear problems have a unique solution, we have proved the existence and uniqueness of the solution of the boundary value problems for the quasillnear elliptic systems.     

ON THE EXISTENCE AND NODAL CHARACTER OF THE SOLUTIONS OF SINGULAR NONLINEAR BOUNDARY VALUE PROBLEMS

§ 1. Introduction We consider the singular nonlinear boundary value problem where l=v+3/v-1,l+1 is the critical exponent of the embedding of weighted Sobolev space W_t~2~(1,2)(O, +∞) into L_t~2~q(O, ∞), v>2. When v=N-1 we can get the radial solutions of problem where 2~*=2N/N-2 is the critical exponent of the Sobolev embedding H~1(R~n)→L~Q(R~N). Kurtz has discussed the existence of κ-node solution of (1.1), (1.2) for each κ∈N U{0} when the growth rate of |u|~(l-1)u+f(u) is lower then |u|~(v+3/v-1) i.e.     

A CLASS OF BOUNDARY PROBLEMS FOR SOME HYPERBOLIC EQUATIONS

A boundary problem for the Klein-Gordon equation in the strip O≤t≤T is considered with the boundary condition:the initial state at t=O and the final state at t=T.It is proven that the problem admits of an infinite number of solutions.The same result holds for a generic 2nd order hyperbolic equation in 2-variables.Using the result for the wave operator in 3-space dimensions we give a method to reconstruct functions whose integral on all unit spheres in R~3 is a given function.     

GLOBAL EXISTENCE OF THE SOLUTION FOR THE LANDAU-LIFSHITZ EQUATION OF THE FERROMAGNETIC SPIN CHAIN

1IntroductionInthepresentpaper,wecollsidertileLandau-LifSllitzequationoftileferromagneticspinchainwhichtakestheform:where"x'ldenotesthevectorcrossproductinR',fi=(--D,D),D>0,u=(ul,u3,u3):fixR -R3istilespinvector,or>0isaGilbertdampingconstant,Pisarealparameter,J=diag(ri,acfac)isaconstantmatrix.TheaboveequationarisesinthedoublyanisotropicHeisenbergspinsystem.Thissystemisimpliedbytileconservationofenergyandmagnitudeofu.SeeDanielandLab.hmanan[1]please.Insomeferromagneticchainsystems,thematri…     

THE PROPERTIES OF MATRICES aI+βJ AND SOME RELATED PROBLEMS

Let F Be a field,v(≥2)an integer,I_v the identity matrix of order v,and J_v the all-one matrix of order v.Let={aI_v+βJ_u,∈GL_u(F)|a,β∈F},and={C∈GL_v(F)|the row-sums and the column-sums of C are all equal}.In this paper,we study (1) the center and the centralizer of ,(2) the centralizer and the normalizer of when v=2,and (3) the orders of and the normalizer of.     

ON THE EXISTENCE AND NODAL CHARACTER OF SOLUTIONS OF SINGULAR NONLINEAR BOUNDARY VALUE PROBLEMS

In this paper we prove the existence of k-node solution of singular nonlinear boundary value problems for each k∈N ∪ {0}.     

THE NATURAL BOUNDARY OF SOME RANDOM POWER SERIES

Suppose that {X_n(ω)} are independent random complex variable sequence, E(X_n)-0 and Then the circle {|Z|=ρ} is almost surely a natural boundary of the random series sum from n=1 to ∞ X_n(ω)Z~(n-1)     

THE WELL-POSEDNESS OF THE ABSTRACT KINETIC EQUATION BOUNDARY VALUE PROBLEMS

In this paper the abstract boundary value problem on Hilbert space H is studied: where Q /Q- are the maximum positive/negative spectral projections of self-adjoint operator T. Under suitable conditions, it is shown that the boundary value problem of abstract kinetic equation is well-posed and can be applied to transport theory.     

THE DEFECT ITERATION OF THE FINITE ELEMENT FOR ELLIPTIC BOUNDARY VALUE PROBLEMS AND PETROV-GALERKIN APPROXIMATION

1.IntroductionFranketc.of.[l]establishedtheiterateddefectcorrectionschemeforfiniteelemelltofellipticboundaryproblems.FOrlinearellipticboundaryvalueproblem[2--5]havediscllssedtheefficiencyoftheschemebyusillgsuperconvergenceandasymptoticexpansion"lidertheco…     

THE COMPARISON OF ASYMPTOTIC BEHAVIOR OF THE SOLUTION OF EQUIVALUED BOUNDARY VALUE PROBLEMS FOR NONLINEAR AND LINEAR ELLIPTIC EQUATIONS

This paper shows the different asymptotic behavior of the solution of equivalued boun-dary value problems for nonlinear equation from the solution to linear one,while the boundary,on which the equivalued boundary value is carried,shrinks to afixed point.     

EXISTENCE OF COUPLED WEAK QUASISOLUTIONS OF SYSTEMS OF QUASILINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS

In this paper, we use the upper-lower solution method with the theory of monotone operators to obtain the existence of coupled weak quasi-solutions of the following boundary value problem for quasilinear elliptic system     

ON THE EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR 2nTH-ORDER BOUNDARY VALUE PROBLEMS

In this paper, by using the Leray-Schauder continuation theorem, we establish the existence and uniqueness theorems of solutions of two-point boundary value problems for 2nth-order nonlinear differential equations with nonlinear growth.     

OSCILLATION OF SYSTEMS OF IMPULSIVE DELAY PARABOLIC EQUATIONS ABOUT BOUNDARY VALUE PROBLEMS

In this paper,oscillation of solutions to a class of impulsive delay para- bolic partial differential equations system with higher order Laplace operator is studied.Under two different boundary value conditions,we establish some sufficient criteria with respect to the oscillations of such systems,employing first-order impulsive delay differential inequalities.The results fully reflect the influence action of impulsive and delay in oscillation.     

THE MODIFIED HELMHOLTZ EQUATIONS AND THEIR BOUNDARY VALUE PROBLEMS

§1. Concept of a Generalized Holomorphic Vector In mathematical problems of space technics, nuclear technics, laser and electron technics, we often encounter many boundary value problems (BVPs) of Modified Helmholtz Equations (MHEs), so how to solve these equations attracts our attention.     

THE DECAY BEHAVIOR OF SOLUTIONS FOR NONLOCAL BOUNDARY PROBLEMS

A parabolic equation defined on a bounded domain with boundary condition being nonlocal is considered. The existence and the dynamic behavior of the solutions for linear and semilinear equations are investigated in special spaces. One will find that the behavior of the solutions are affected by the boundary conditions. A semigroup approach is employed in this paper.     

A FORMULA OF THE KINEMATIC MEASURE IN NONEUCLIDEAN SPACE AND SOME PROBLEMS ON THE GEOMETRIC PROBABILITIES

Let P_n, be the n-dimensional projective space. Let x_0,x_1,…,x_n be the homogeneous coordinates of point x. Consider the quadric φ(x) =x_0~2+x_1~2+…x_(n-1)~2+εx_n~2, where ε can take the value of +1 or -1. We assume the homogeneous coordinates of the points x that do not belong to φ = 0 are normalized, such that     

PERIODIC BOUNDARY PROBLEMS FOR SEMI-LINEAR DEGENERATE EVOLUTION SYSTEMS OF HIGHER ORDER

The semi-linear partial differential systems are often found in physics,chemistry reaction, biology, etc. In this paper we disscuss the existence, uniqueness in global and regularities for a class of semi-linear systems with derivatives of higher order as follows:     

ON BOUNDARY VALUE PROBLEMS FOROVERDETERMINED ELLIPTIC SYSTEMS OF TWO COMPLEX VARIABLES

In this paper,based on previous results,the Riemann-Hilbert boundary valueproblems with general forms for overdetermined elliptic equations of first order are consi-dered.The characteristic of modified function space is given.It is proved that there existsa unique solution for modified problem of the problem which we discuss.By the way,itis pointed out that there are great differences between overdetermined elliptic systems andfirst order elliptei systems in the plane.