THE ASYMPTOTIC THEORY OF INITIAL VALUE PROBLEMS FOR SEMILINEAR PERTURBED WAVE EQUATIONS IN TWO SPACE DIMENSIONS

IntroductionInthispaper,anasymptotictheoryisestablishedforthefollowinginitialvalueproblemforasemilinearperturbedwaveequation :utt-Δu=εf(u ,ε)　　 (t>0 ,x∈R2 ) ,(1 )u(0 ,x,ε) =u0 (x ,ε)　　 (x∈R2 ) ,(2 )ut(0 ,x,ε) =u1(x ,ε)　　 (x∈R2 ) ,(3 )where     

Asymptotic theory of initial value problems for nonlinear perturbed Klein-Gordon equations

Introduction InthispaperasymptotictheoryofthefollowinginitialvalueproblemforanonlinearKlein Gordonequationisconsidered.tt-Δ =εF(t,x,,ε),t>0,x∈R2,(0,x,ε)=0(x,ε),t(0,x,ε)=1(x,ε),x∈R2,(1)where(t,x)isarealvaluedunknownfunction,Δ=2i     

On the isometric isomorphism of probabilistic metric spaces

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｒｅａｒｅｔｗｏｋｉｎｄｓｏｆｉｓｏｍｅｔｒｉｃｉｓｏｍｏｒｐｈｉｓｍｉｎｐｒｏｂａｂｉｌｉｓｔｉｃｍｅｔｒｉｃｓｐａｃｅｔｈｅｏｒｙ (ｓｅｅ[1 -9] ) .ＴｈｅｆｉｒｓｔｉｓｔｈａｔａＰＭｓｐａｃｅ (Ｅ ,Ｆ)ｉｓｉｓｏｍｅｔｒｉｃａｌｌｙｉｓｏｍｏｒｐｈｉｃｔｏａｎｏｔｈｅｒＰＭｓｐａｃｅ(Ｅ′,Ｆ′) ,ａｎｄｔｈｅｓｅｃｏｎｄｉｓｔｈａｔａＰＭｓｐａｃｅｓ(Ｅ ,Ｆ)ｉｓｉｓｏｍｅｔｒｉｃａｌｌｙｉｓｏｍｏｒｐｈｉｃｔｏａｇｅｎｅｒａｔｉｎｇｓｐａｃｅｏｆｑｕａｓｉ＿ｍｅｔ…     

ON THE WELL POSEDNESS OF INITIAL VALUE PROBLEM FOR EULER EQUATIONS OF INCOMPRESSIBLE INVISCID FLUID(Ⅰ)

The ill posed initial value problem of the Euler equations and the formal solvability of ill posed problem based on stratification theory are discussed. For some ill posed initial value problems, the existence conditions of formal solutions and the methods of how to construct a formal solution are given. Finally, an example is given to discuss the ill posedness of the initial value problem on hyper plane {t=0} in R⁴, and explain that the problem has more than one solution. Foundation item: the National Natural Science Foundation of China (19971054) Biography: Shen Zhen (1977−)     

The asymptotic theory of semilinear perturbed telegraph equation and its application

I.IntroductionConsiderthefollowingsemilinearperturbedtelegraphequationuII-u., P'u==sj(t,x,u,ul,u,,e)(-ooo)(l.l)u(o,x)=u,(x,e)(-ooo,u=u(t,x),fuoandulsatisfycertainsmoo…     

ON THE WELL POSEDNESS OF INITIAL VALUE PROBLEM FOR EULER EQUATIONS OF INCOMPRESSIBLE INVISCID FLUID(Ⅱ)

The solvability of the Euler equations about incompressible inviscid fluid based on the stratification theory is discussed. And the conditions for the existence of formal solutions and the methods are presented for calculating all kinds of ill-posed initial value problems. Two examples are given as the evidences that the initial problems at the hyper surface does not exist any unique solution. Foundation item: the National Natural Science Foundation of China (19971054) Biography: Shen Zhen (1977−)     

The nonlinear nonlocal singularly perturbed problems for reaction diffusion equations

A class of nonlinear nonlocal for singularly perturbed Robin initial boundary value problems for reaction diffusion equations is considered. Under suitable conditions, firstly, the outer solution of the original problem is obtained, secondly, using the stretched variable, the composing expansion method and the expanding theory of power series the initial layer is constructed, finally, using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems are studied and educing some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation is discussed. Foundation items: the National Natural Science Foundation of China (10071048); the “Hunfred Talents Project” by Chinese Academy of Sciences Biography: Mo Jia-qi (1937−)     

The thermal response of a metal slab to a class of radially dependent heat inputs

Summary The temperature fields are deduced for an infinite metal slab of finite thickness produced by a heat flux input which depends on the radial coordinate. The formal solution is obtained by means of transform techniques.Several representations of the Green's function are obtained as well as two approximate representations. One of these approximations converges well for time large the other for time small.A small time asymptotic expansion is given for input flux densities which satisfy certain smoothness requirements.Two inputs have been considered in detail. One is the uniform spot and the other is the Gaussian. An extension of the results obtained by Oosterkamp³) has been obtained along with an estimate of the validity.The work reported in this paper was supported in part by Contract AF 30(602)-2305 between The Ohio State University Research Foundation and Rome Air Development Center, Griffiss Air Force Base, New York.     

TheC k instability of navier-Stokes equation appending polynomials of unknown functions

Applying the theory of stratification, the solution space structure about a class of deformed Navier-Stokes equation isdetermined. It is proved that such kind of equation has no C^k (k≥2) stable solution by the fact that the strate transversale is a null set. Foundation item: the National Natural Science Foundation of China (19971054), Shanghai Natural Science Foundation (99ZA14034) Biography: HE You-hua (1960-)     

Some problems in the Z-C-X space

The new concepts of the Z-C-X space and excellent cone are introduced. Some problems of random semiclosed 1-set-contractive operator are investigated in the Z-C-X space. At first, an important inequality is proved. Secondly, several new conclusions are proved by means of random fixed point index in the theory of random topological degree. A random solution of a class of random operator equations under conditions of imitating the parallelogram law is obtained, famous Altman’s theorem is generalized in partially ordered Z-C-X space. Therefore, some new results are obtained.     

An analytic and application to state space reconstruction about chaotic time series

The state space reconstruction is the major important quantitative index for describing non-linear chaotic time series. Based on the work of many scholars, such as: N.H.Packard, F.Takens, M. Casdagli, J.F.Gibson, CHEN Yu-shu et al, the state space was reconstructed using the method of Legendre coordinate. Several different scaling regimes for lag time τ were identified. The influence for state space reconstruction of lag time τ was discussed. The result tells us that is a good practical method for state space reconstruction.     

ON THE MAXIMAL LYAPUNOV EXPONENT FOR A REAL NOISE PARAMETRICALLY EXCITED CO-DIMENSION TWO BIFURCATION SYSTEM (Ⅰ)

For a real noise parametrically excited co-dimension two bifurcation system on three-dimensional central manifold, a model of enhanced generality is developed in the present paper by assuming the real noise to be an output of a linear filter system, namely, a zero-mean stationary Gaussian diffusion process that satisfies the detailed balance condition. On such basis, asymptotic expansions of invariant measure and maximal Lyapunov exponent for the relevant system are established by use of Arnold asymptotic analysis approach in parallel with the eigenvalue spectrum of Fokker-Planck operator. Foundation item: the National Natural Science Foundation of China (19602016)     

On the formal solution of initial value problem of Navier-Stokes equation

Ｉｎｒｅｆｅｒｅｎｃｅ [1 ] ,ｗｅｄｉｓｃｕｓｓｅｄｔｈｅｉｎｓｔａｂｉｌｉｔｙｏｆＮａｖｉｅｒ＿ＳｔｏｋｅｓｅｑｕａｔｉｏｎｏｎＲ3×Ｒ ,ａｎｄ ,ｉｎｔｈｅｉｌｌｕｓｔｒａｔｉｏｎｏｆｉｔ,ｗｅｐｒｅｓｅｎｔｅｄｓｏｍｅｅｘａｍｐｌｅｓｉｎｔｈｅｆｏｒｍｏｆｅｘａｃｔｓｏｌｕｔｉｏｎ .Ｔｈｅｂａｓｉｓｏｆｔｈｅｃｏｎｓｔｒｕｃｔｉｏｎｏｆｔｈｅｍｉｓｔｈｅ“ｅｑｕａｔｉｏｎｓｅｃｏｎｄａｉｒｅ”ｏｆｔｈｅｅｑｕａｔｉｏｎ .Ｉｎｆａｃｔ,ｔｈｅｓｅｅｘａｃｔｓｏｌｕｔｉｏｎｓａｒｅｓｏｍｅｌｉｍｉｔ…     

ON THE MAXIMAL LYAPUNOV EXPONENT FOR A REAL NOISE PARAMETRICALLY EXCITED CO-DIMENSION TWO BIFURCATION SYSTEM (Ⅱ)

For a co-dimension two bifurcation system on a three-dimensional central manifold, which is parametrically excited by a real noise, a rather general model is obtained by assuming that the real noise is an output of a linear filter system-a zeromean stationary Gaussian diffusion process which satisfies detailed balance condition. By means of the asymptotic analysis approach given by L. Arnold and the expression of the eigenvalue spectrum of Fokker-Planck operator, the asymptotic expansions of invariant measure and maximal Lyapunov exponent for the relevant system are obtained. Foundation item: the National Natural Science Foundation of China (19602016)     

MICROMECHANICS ANALYSIS ON EVOLUTION OF CRACK IN VISCOELASTIC MATERIALS

ＩｎｔｒｏｄｕｃｔｉｏｎＡｓｓｈｏｗｎｂｙＺＨＡＮＧａｎｄＸＩＯＮＧ ( 1 997) [1],ｕｎｄｅｒｃｏｎｓｔａｎｔｌｏａｄｉｎｇｔｈｅｃｒａｃｋｉｎｖｉｓｃｏｅｌａｓｔｉｃｍａｔｅｒｉａｌｓｈａｓａｎｉｎｃｕｂａｔｉｏｎｔｉｍｅ ,ｗｉｔｈｉｎｉｔｔｈｅｃｒａｃｋｏｐｅｎｉｎｇｄｉｓｐｌａｃｅｍｅｎｔ (ＣＯＤ)ｉｓｃｈａｎｇｉｎｇｌａｒｇｅｒｗｉｔｈｔｉｍｅ,ｗｈｉｌｅｔｈｅｃｒａｃｋｌｅｎｇｔｈｋｅｅｐｓｃｏｎｓｔａｎｔ.Ｉｎｖｅｓｔｉｇａｔｉｏｎｏｎｔｈｅｇｒａｄｕａｌｏｐｅｎｉｎｇｏｆｔｈｅｃｒａｃ…     

Noether's theory of mechanical systems with unilateral constraints

Ｎｏｅｔｈｅｒ’ｓｔｈｅｏｒｅｍｒｅｖｅａｌｓｔｈｅｉｎｎｅｒｃｏｎｎｅｃｔｉｏｎｂｅｔｗｅｅｎｔｈｅｃｏｎｓｅｒｖａｔｉｏｎｌａｗｓａｎｄｔｈｅｄｙｎａｍｉｃａｌｓｙｍｍｅｔｒｙｏｆｄｙｎａｍｉｃａｌｓｙｓｔｅｍｓ．Ｉｎｔｈｅｒｅｃｅｎｔｔｗｅｎｔ...     

Conservation Law and Application of J-Integral in Multi-Materials

The conservation law of J-integral in two-media with a crack paralleling to the interface of the two media was firstly proved by analytical and numerical finite element method. Then a schedule model was established that an interface crack is inserted in four media. According to the J-integral conservation law on multi-media, the energy release ratio of Ⅰ-type crack was considered to be conservation when the middle medium layers are very thin. And the conservation law was also convinced by numerical method. By means of the dimension analysis on the model, the asymptotic results and formula calculating the energy release ratio and complex stress intensity factor are presented.     

Uniform Analytic Construction of Wavelet Analysis Filters Based on Sine and Cosine Trigonometric Functions

1　Ｉｎｔｒｏｄｕｃｔｉｏｎ1 .1ＩｎｔｒｏｄｕｃｔｉｏｎｏｆｂａｃｋｇｒｏｕｎｄＷａｖｅｌｅｔａｎａｌｙｓｉｓａｎｄｉｔｓａｐｐｌｉｃａｔｉｏｎｓｈａｖｅｂｅｃｏｍｅｏｎｅｏｆｔｈｅｆａｓｔｅｓｔｇｒｏｗｉｎｇｒｅｓｅａｒｃｈａｒｅａｓｉｎｒｅｃｅｎｔｙｅａｒｓ[1- 9].Ｅｓｐｅｃｉａｌｌｙ ,ｔｈｅｗａｖｅｌｅｔｔｒａｎｓｆｏｒｍｓｂａｓｅｄｏｎｏｒｔｈｏｎｏｒｍａｌｂａｓｅｓｏｆｃｏｍｐａｃｔｌｙｓｕｐｐｏｒｔｅｄｗａｖｅｌｅｔａｒｅｗｉｄｅｌｙｕｓｅｄｉｎｉｍａｇｅｃｏｍｐｒｅｓｓｉｏｎ ,ｃｏｍ…     

On the stability boundary of hamiltonian systems

The criterion for the points in the parameter space being on the stability boundary of linear Hamiltonian system depending on arbitrary numbers of parameters was given, through the sensitivity analysis of eigenvalues and eigenvectors. The results show that multiple eigenvalues with Jordan chain take a very important role in the stability of Hamiltonian systems. Foundation item: the National Natural Science Foundation of China (10072012); the National Natural Science Foundation of Russia Biography: QI Zhao-hui (1964-)     

D'alembert principle in the velocity space

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｒｅｃｅｎｔｙｅａｒｓ,ｒｅｓｅａｒｃｈｅｒｓｈａｖｅｂｅｅｎｍａｄｅｏｎｔｈｅｖａｒｉｏｕｓｆｏｒｍｕｌａｅｏｆＤ’Ａｌｅｍｂｅｒｔｐｒｉｎｃｉｐｌｅ,ａｔｔｅｎｔｉｏｎｈａｓｂｅｅｎｇｉｖｅｎｔｏｍｅｃｈａｎｉｃａｌｐｒｉｎｃｉｐｌｅｓｉｎｔｈｅｖｅｌｏｃｉｔｙｓｐａｃｅａｎｄｔｈｅｖａｒｉａｔｉｏｎａｌｐｒｉｎｃｉｐｌｅｉｎｔｈｅｖｅｌｏｃｉｔｙｓｐａｃｅ,ａｎｄｔｈｅｖｅｌｏｃｉｔｙｓｐａｃｅｏｆｒｅｌａｔｉｖｉｓｔｉｃｍｅｃｈａｎｉｃｓｈａｓｂｅｅｎｍａｄｅ…