关于微分方程的一个混合边界问题

F. G. Tricomi (1923- ), S. Gellerstedt (1935- ), F. I . Frankl (1945- ), A. V. Bitsadze and M. A. Lavrentiev (1950- ), M. H. Protter (1953- ) and most of the recent workers in the field of mixed type boundary value problems have considered only one parabolic line of degeneracy. The problem with more than one parabolic line of degeneracy becomes more complicated. The above researchers and many others have restricted their attention to the Chaplygin equation:K(y)·u_xx+u_yy =f(x,y) and not considered the "generalized Chaplygin equation:"Lu=K(y)·u_xx+u_yy+r(x,y)·u=f(x,y) because of the difficulties that arise when r₁=non-trivial(≠0). Also it is unusual for anyone to study such problems in a doubly connected region. In this paper 1 consider a case of this type with two parabolic lines of degeneracy, r₂=non-trivial(≠0).in a doubly connected region, and such that boundary conditions are prescribed only on the-exterior boundary" of the mixed domain, and 1 obtain umqueness results for quasnegutar solutions of the characteristic and non-characteristic Problem by applying the b,c energy integral method in the mixed domain.     

一类非线性混合型方程的Tricomi问题

本文考虑非线性混合型方程 k(x,y)u_xx+u_yy+α(x,y)u_x+β(x,y)u_y+γ(x,y)u-|u|^ρu=f(x,y)的Tricomi 问题.利用能量积分和不动点原理,在很弱的条件下证明了H¹强解的存在性.     

几个非线性演化方程的解析解

本文我们求出了K—P方程u_xt+6(uu_x)_x+u_xxxx+3k²u_yy=0和Boussinesq方程u_tt-u_xt-6(u²)_xx+u_xxxx=0的孤立波解族.求出了广义Schr?dinger方程iu_t+u_xx-u相似文献   

Long-time Asymptotic Behavior for the Derivative Schr¨odinger Equation with Finite Density Type Initial Data*

In this paper, the authors apply ? steepest descent method to study the Cauchy problem for the derivative nonlinear Schr¨odinger equation with finite density type initial data iq_t + q_xx + i(|q|²q)x = 0,q(x, 0) = q0(x),where lim/x→±∞ q0(x) = q± and |q±| = 1. Based on the spectral analysis of the Lax pair,they express the solution of the derivative Schr¨odinger equation in terms of solutions of a Riemann-Hilbert problem. They compute the long time asymptotic expansion of the solution q(x, t) in different space-time regions. For the region ξ =x/t with |ξ + 2| < 1, the long time asymptotic is given by q(x, t) = T (∞)^?2q^r_Λ(x, t) + O(t^?3/4 ),in which the leading term is N(I) solitons, the second term is a residual error from a ? equation. For the region |ξ + 2| > 1, the long time asymptotic is given by q(x, t) = T (∞)^?2q^r_Λ(x, t) ? t^?1/2 if¹¹ + O(t^?3/4 ),in which the leading term is N(I) solitons, the second t^?1/2 order term is soliton-radiation interactions and the third term is a residual error from a ? equation. These results are verification of the soliton resolution conjecture for the derivative Schr¨odinger equation. In their case of finite density type initial data, the phase function θ(z) is more complicated that in finite mass initial data. Moreover, two triangular decompositions of the jump matrix are used to open jump lines on the whole real axis and imaginary axis, respectively.     

形态学上一个非线性偏微分方程的正定性及渐近解

One reaction-diffusion equation φ_tx+εφ_tφ_xx-φ_xxx=γ(φ_x) has been presented in the study of Morphogenesis. In this paper, reasonable de-finite conditions of the equation are proposed and the asymptotic form of its solution is obtained by using perturbation method. So the existence of solution of this probiem is solved.     

Local Well-posedness of the Derivative Schr¨odinger Equation in Higher Dimension for Any Large Data

In this paper, the authors consider the local well-posedness for the derivative Schr¨odinger equation in higher dimension u_t ? i△u + |u|²(?→γ · ?u) + u²(?→λ · ?u) = 0, (x, t) ∈ Rⁿ × R,?→γ ,?→λ ∈ Rⁿ; n ≥ 2.It is shown that the Cauchy problem of the derivative Schr¨odinger equation in higher dimension is locally well-posed in H^s(Rⁿ) (s > n/2) for any large initial data. Thus this result can compare with that in one dimension except for the endpoint space H^n/2.     

混合曲率曲面变形问题的Tricomi问题

本文研究由混合曲率曲面无穷小变形所得的混合型方程rw_yy-2sw_xy+tw_xx=0的Tricomi问题。在曲面满足某些条件下,利用能量积分法证明了对这个方程的Tricomi问题的唯一性,所提出的条件对(?)方程的特殊情形和(?),Protter等人所得的条件相一致。     

On a nonhomogeneous Burgers’ equation

In this paper the Cauchy problem for the following nonhomogeneous Burgers’ equation is considered : (1)u _t +uu _x =μu _xx −kx,x ∈R,t > 0, where μ and k are positive constants. Since the nonhomogeneous term kx does not belong to any L^p(R) space, this type of equation is beyond usual Sobolev framework in some sense. By Hopf-Cole transformation, (1) takes the form (2)ϕ _t −ϕ _xx = −x ² ϕ. With the help of the Hermite polynomials and their properties, (1) and (2) are solved exactly. Moreover, the large time behavior of the solutions is also considered, similar to the discussion in Hopf’s paper. Especially, we observe that the nonhomogeneous Burgers’ equation (1) is nonlinearly unstable.     

The Mixed Bitsadze-Lavrent'ev-Tricomi Boundary Value Froblem

It is well known that F.G. Tricomi (1923) is the originator of the theory of boundary value problems for mixed type equations by establishing the Thicomi equation: y. u_(xx) u_(yy)=0 which is hyperbolic for y<0, elliptic for y(?)0, and parabolic for y=0 and then applied it in the theory of transonic flows.     

The Exterior Tricomi and Frankl Problem

F. G. Tricomi (1923—), S. Gellerstedt (1935—), F.I.Frankl (1945—),A. V. Bitsadze and M. A. Lavrentiev (1950—), M. H. Protter (1953—) and most of the recent workers in the field of mixed type boundary value problems have considered only one parabolic line of degeneracy. The problem with more than one parabolic line of degeneracy becomes more complicated. The above researchers and many others have restricted their attention to the Chaplygin equation:K(y)·u_(xx)+u_(yy)=f(x, y) and not considered the "generalized Chaplygin equation:"Lu=K(y)·u_(xx)+u_(yy)+r(x, y)·u=f(x, y) because of the difficulties that arise when r:=non-trivial (≠0). Also it is unusual for anyone to study such problems in a doubly connected region. In this paper 1 consider a case of this type with two parabolic lines of degeneracy, r:= non-(?)≠(?), in a doubly connected region,and such that boundary conditions are presenbed only on the "exterior boundary" of the mixed domain, and Ⅰobtam uniqueness (?) for quasllegular solutions     

半质环交换性的一点注记

Let R be a semi-prime ring, C be the center of R. Let F_i (x, y) (i = 1, 2) be a product of the m times x's and n times y's.In this paper following theorem is proved: (I ) implies (Ⅱ), where( Ⅰ )If f₁(x,y) -f₂(x,y) ∈C for every x,y in R, then R is commutative;(Ⅱ)If f₁ (x,y) + f₂(x, y) ∈C for every x,y in R, then R is commutative.Thus very short proves of some theorems of references[5], [8], [9] are be given.     

Positive Solutions for Asymptotically Linear Cone-Degenerate Elliptic Equations

In this paper, the authors study the asymptotically linear elliptic equation on manifold with conical singularities ??_Bu + λu = a(z)f(u), u ≥ 0 in R^N₊,where N = n + 1 ≥ 3, λ > 0, z = t,x₁,· · · ,x_n, and ?_B = (t?_t)^{2 + ?2_x1 + · · · + ?2_xn. Combining properties of cone-degenerate operator, the Pohozaev manifold and qualitative properties of the ground state solution for the limit equation, we obtain a positive solution under some suitable conditions on a and f.}     

Stability of Rotation Relations of Three Unitaries with the Flip Action in C*-Algebras*

The authors show that if Θ = (θ_jk) is a 3 × 3 totally irrational real skewsymmetric matrix, where θ_jk ∈ [0, 1) for j, k = 1, 2, 3, then for any ε > 0, there exists δ > 0 satisfying the following: For any unital C^*-algebra A with the cancellation property,strict comparison and nonempty tracial state space, any four unitaries u₁, u₂, u₃, w ∈ A such that (1) u_ku_j - e^{2πiθjk ujukk} < δ, wu_jw^-1 = u^-1_j, w² = 1_A for j, k = 1, 2, 3; (2)τ (aw) = 0 and τ ((u_ku_ju_k^*u_j^* )ⁿ ) = e^2πinθjk for all n ∈ N, all a ∈ C^*(u₁, u₂, u₃), j, k = 1, 2, 3 and all tracial states τ on A, where C^*(u₁, u₂, u₃) is the C^*-subalgebra generated by u₁, u₂ and u₃, there exists a 4-tuple of unitaries u₁, u₂, u₃, w in A such that u_ku_j = e^2πinθjk u_ku_j, w u_j w^-1 = u^-1 _j, w² = 1A and k u_j - u_jk < ε, k w - wk < ε for j, k = 1, 2, 3. The above conclusion is also called that the rotation relations of three unitaries with the flip action is stable under the above conditions.     

二值回归模型中自变量误差的处理

In this paper the binary regression model, in which the response Y is binary, i.e. Y takes the value 0 or 1 only, is considered. The conditional probability P{Y=1|X=x} is assumed to be of the form G(a₀+a₁x), where x is the measurement of the factor, a= (a₀,a₁)′ is the vector of the unknown parameter being estimated and G(a) = (1+e^a)^-1. The measurement x is assu-med to be contaminated when the value of x is out of certain limit. M-estimation is employed to estimate the unknown a of the model. In this paper, the consistent M-estimation is proved to be asymptotically normal and the optimal M-equation is obtained. Finally, a computation method is introduced to solve the M-equation and to get the optimal solution.     

Distance Between Unitary Orbits of Self-AdjointElements in C*-Algebras of Tracial Rank One

The note studies certain distance between unitary orbits. A result about Riesz interpolation property is proved in the first place. Weyl (1912) shows that dist(U(x), U(y)) = δ(x, y) for self-adjoint elements in matrixes. The author generalizes the result to C^*-algebras of tracial rank one. It is proved that dist(U(x), U(y)) = D_c(x, y) in unital AT -algebras and in unital simple C^*-algebras of tracial rank one, where x, y are self-adjoint elements and D_c (x, y) is a notion generalized from δ(x, y).     

关于Hammerstein型非线性积分方程固有值的一点注记

Let Aφ(x)=∫_GK(x,y)f(y,φ(y))dy, where G is a bounded closed domain in Euclidean space, K(x,y) is continuous on G×G, f(x,u) is continuous on G×R, and f(x,0)≡0. Set G_x={x|x∈G,K(x,y)≠0},G_y={y|y∈G,K(x,y)≠0},G₁=G_x∩G_y≠φ.Let K₁(x,y) be the restriction of K(x,y) on G₁×G₁,f₁(x,u)be the restriction of f(x, u) on G₁× R, and A₁φ=∫_G1K₁(x,y)f₁(y,φ₁(y))dy, The main result of this paper is Theorem λ≠0 is an eigenvalue of A, if and only if λ is an eigenvalue of A₁.     

一个重特征方程的一类定解问题

本文提出了当p=2h+1(h≥0为整数)时,重特征方程 L_pu≡u_xx-x²u_ii+pu_i=0 的一类新的定解问题,证明其适定性,推广了[2]、[3]和[4]中的相应结果。     

关于Dafermos折线逼近法的注记

The remark in [1] shows that the polygonal approximations suggests the numerical method of construction of a solution of the initial value problem of a conservation law for piecewise linear f(u) and u₀(x) a step function. Unfortunately, there is no guarantee that one can reach by this procedure every point t∈[0,∞) in a finite number of steps. In the present note, we give this guarantee.     

关于具有多项式约束的I-环

In this paper, it is shown that an l-prime lattice-ordered ring in which the square of every element is positive must be a domain provided it has non-zero f-elements and be an l-domain provided it has a left (right) identity ele-ment or a central idempotent element .More generally,the same conclusion follows if the condition a²≥0 is replaced by p(a)≥0 or f(a,b)≥0 for suitable polyno-mials p(x) and f(x, y) . It is also shown that an l-algebra is an f-algebra provided it is archimedean, contains an f-element e>0 with r₁(e)=0, and satifies a polynomial identity p(x)≥0 or f(x,y)≥0 (for suitable f(x,y)).     

质环的求导和交换性

In this paper, we generalize some corresponding results of [1-4]. We obtain the main results as the following:Theorem 1 Let R be a prime ring of characteristic not 2 with nontrivial derivations d₁,d₂ and let U be a nonzero ideal of R . If C is the center of R, then the following conditions are equivalent : (i)d₁,d₂(x)∈C for all x∈U; (ii) [d₁(x),d₂(y)]∈C for all x,y∈U; (iii) d₁(x)d₂(y)+d₂(x)d₁(y)∈C for all x,y∈U; (iv) R is commutative.Theorem 2 Let R be a prime ring with nontrivial derivations d₁,d₂,…, d_n and U be a nonzero ideal of R. Let C be the center of R. If d₁(x₁)d₂(x₂)…d_n(x_n)∈C for all x₁, x₂…x_n)∈U, then R is commutative.