Über die Differentialgleichungu x y =f(x,y, u,u x,u y )

Ohne ZusammenfassungDiese Arbeit wurde teilweise unterstützt von der United States Air Force [Air Force Office of Scientific Research-Contract AF 49 (638)-228]. Sie besteht aus drei Teilen. Teil I ist im vorangehenden Heft dieser Zeitschrift erschienen, Teil III erscheint im folgenden Heft. Formeln und Sätze sind in jedem Teil für sich durchnumeriert. Bei Verweisen bedeutet z. B. (I. 17) die Formel (17) von Teil I, Satz I. 5 bedeutet Satz 5 von Teil I.     

Über die Differentialgleichungu xy =f(x,y, u,u x,u y )

Ohne ZusammenfassungDiese Arbeit wurde teilweise unterstützt von der United States Air Force [Air Force Office of Scientific Research-Contract AF 49(638)-228]. Sie wurde von der Fakultät für Natur- und Geisteswissenschaften der Technischen Hochschule Karlsruhe als Habilitationsschrift angenommen. Die beiden ersten Teile sind in Band 71 dieser Zeitschrift erschienen (S. 308–324 und 436–453).     

Über die Differentialgleichungu xy =f(x,y, u,u x,u y )

Ohne ZusammenfassungDiese Arbeit wurde teilweise unterstützt von der United States Air Force (Air Force Office of Scientific Research-Contract AF 49 (638)-228). Sie besteht aus insgesamt drei Teilen. Teil II (Existenzsätze für die charakteristische Anfangswertaufgabe) und Teil III (Die nichtcharakteristische Anfangswertaufgabe) werden in Kürze in dieser Zeitschrift erscheinen.     

On the Cauchy problem for D t 2–D x a(t,x) n D x

Abstract The well posedness of the Cauchy problem for the operator P=D_t²–D_xa(t,x)ⁿD_x, with data on t=0 is studied assuming a ∈ C^N( (R)), s₀>1 and sufficiently close to 1, a(t,x)≥ 0. Well posedness is proved in Gevrey classes γ^(s), for , n≥ n₀. Keywords: Partial differential equations, Cauchy problem, Well posedness     

The Existence of a Positive Solution of D α[x(t) − x(0)] = x(t)f(t,x t )

This paper is devoted to the problem of existence of positive solution for a delay differential equation with fractional order. Some sufficient conditions for its existence of positive solution are given. This work was supported in part by the Key Project of the National Nature Science Foundation of China (No. 60534020), the National Nature Science Foundation of China (No. 60474037), and Program for New Century Excellent Talents in University (No. NCET-04-415).     

高维半线性波动方程□u=f(t,x,u,Du)解的奇性分析

本文研究半线性波动方程□u=f(t,x,u,Du)解的自蔓。与M.Beals和刘林启的工作比较([7,8]),本文从不同的空间出发,得到了与[7,8]相同的结果。     

The integrable nonlinear degenerate diffusion equationu t=[f(u)u x −1 x and its relatives

The nonlinear diffusion equationu _t=[f(u)g(u _x )] arises in recent models of turbulent transport and of stress dissipation in rock blasting. A Lie point symmetry analysis produces many similarity reductions of exponential and power-law forms, and reveals that for all choices off the equation is always integrable wheng(u _x )=1/u _x . We identify the functionsf(u) which guarantee equivalence to the linear heat equation. For all other choices off, the linear canonical form leads to a self-adjoint differential equation by separation of variablesx andt. We construct a number of explicit solutions with simple boundary conditions, which illustrate behavior in the vicinity of the degenerate region withu _x =. If zero flux and constant concentration are maintained on free boundaries, then steep concentration gradients may evolve from smooth initial conditions. For other boundary conditions, unlike the examples of strong degeneracy, smoothing will occur at initial step discontinuities.     

When isf(x,y) = u(x) + v(y)?

LetX andY be arbitrary non-empty sets and letS a non-empty subset ofX ×Y. We give necessary and sufficient conditions onS which ensure that every real valued function onS is the sum of a function onX and a function onY.     

THE STABILITY OF RDDE (?)(t) p_1(t)x(t) q_1(t)x(t) p_2(t)x(t-r(t)) q_2(t)x(t-r(t))=0

In this paper, the author considered the stability of zero solution of linear RDDEx(t) p_1(t)x(t) q_1(t)x(t) p_2(t)x(t-r(t)) q_2(t)x(t-r(t))=0,(1)x(t) p_1(t)x(t) q_1(t)x(t) q_2(t)x(t-r(t))=0 (2)using Liapunov-Razumikhin functional and transformations and obtained some sufficient condi-tions for the stability of Eqs.(1) and (2). These results are suitable both for bounded p_i(t), q_i(t)and r(t).i =1, 2.     

Generalized (f,U s,u t , λ s t )-structures

iauliai Pedagogical Institute, P. Viinskio str. 25, 5419 iauliai, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 33, No. 2, pp. 135–141, April–June, 1993.     

Asymptotic behavior for t→∞ of the solutions of the equation ψxx+u(x,t)ψ+(λ/4)ψ=0 with a potential u,satisfying the Korteweg-de Vries equation. II

This is the third paper in a series of papers of the authors, devoted to a rigorous investigation of the asymptotic behavior of the solutions of the KdV equation as t. The immediate purpose of the paper is the investigation of the solution of the Schrödinger equation in the neighborhood of the singular point x=3t for a special class of potentials, introduced in the previous papers. As it will be proved, in the final analysis this class of potentials describes the asymptotic behavior of the solutions, decreasing for x, of the KdV equation as t. The solution , far from the singular point, has been investigated earlier. In the paper we investigate a series which gives the solution of the Schrödinger equation and we consider the asymptotic properties of this series.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 138, pp. 8–32, 1984.     

A modification of Newton's method to solve the Dirichlet problem for the equation Δu=f(x,u)

The Dirichlet problem for a weakly nonlinear equation u=f(x, u) is investigated. We use successive approximations constructed by modified Newton's scheme and apply the extremal properties of the solutions of the elliptic equation of the form u – c(x)u=F(x), where c(x) 0. Numerical solution of the resulting sequence of linear boundary-value problems is considered.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 61, pp. 27–30, 1987.     

Asymptotic Behavior of Solutions of the Cauchy Problem x′ = f(t,x, x′), x(0) = 0

We prove the existence of continuously differentiable solutions such that

二阶 Hamilton 系统ü-L(t)u+W′u(t,u)=O同宿轨道的存在性

本文利用临界点理论中的山路引理,证明了二阶 Hamilton 系统ü-L(t)u+W1u(t,u)=0存在非平凡的同宿轨道,其中L(t)y.y≥λy2,y∈Rn,λ＞0,L(t)和W(t,u)关于变量t没有周期性假设.     

从u(x,y)+iu(x,y)到f(z)的一种转化法

<正> 在复变量函数的运算中,常常会遇到需要把以二元函数u(x,y)、v(x,y)为实部与虚都构成的复变量函数f(x+iy)=u(x,y)+iv(x,y)化成以复数z=x+iy为变量的函数f(z),一般常用的方法是:     

方程组x=f(t,x)解的一致稳定性

本文给出了方程组x=f(t,x)解的一致稳定性的一个判别准则,改进了文献[1]中的有关结果。     

Hölder Estimates of Solutions on the Equation u t = Δ G ( u )

In this paper we are concerned with the Hölder estimates of solutions of the Cauchy problem for the degenerate parabolic Eq. (1) with the initial data (2), where the diffusion function G ( u ) can be a constant on a non-zero measure set, such as the equations of two-phase Stefan's type. Under the condition | GG "/ G ' 2 | h g , g 2 h 1/2 N , the global estimates | o ( G f ( u ))| h M , on R N ‐ R + , is obtained by using the maximum principle, where f is a constant given in (9).     

Group Laws [x,y −1] ≡ u(x,y) and Varietal Properties

Let F = ?x, y? be a free group. It is known that the commutator [x, y ^?1] cannot be expressed in terms of basic commutators, in particular in terms of Engel commutators. We show that the laws imposing such an expression define specific varietal properties. For a property 𝒫 we consider a subset U(𝒫) ? F such that every law of the form [x, y ^?1] ≡ u, u ∈ U(𝒫) provides the varietal property 𝒫. For example, we show that each subnormal subgroup is normal in every group of a variety 𝔙 if and only if 𝔙 satisfies a law of the form [x, y ^?1] ≡ u, where u ∈ [F′, ?x?].