Banach空间中非光滑算子方程的光滑化拟牛顿法

研究Banach空间中非光滑算子方程的光滑化拟牛顿法.构造光滑算子逼近非光滑算子,在光滑逼近算子满足方向可微相容性的条件下,证明了光滑化拟牛顿法具有局部超线性收敛性质.应用说明了算法的有效性.     

一阶非线性方程解的存在性问题

<正> 1.引言.本文考虑一阶非线性方程的 Cauchy 问题:(?)古典理论证明,如果 F 关于其变元有连续的二阶偏导数,且 u_0(x)的二阶导数连续,则在初始节 t=0(x_1≤x≤x_2)的某个邻域中,Cauchy 问题(1.1),(1.2)存在唯一的二阶偏导数连续的解.这个解是由特征理论构造的.1928年,A.Haar 在[1]中曾证明了上述问题在 C~1类内解的唯一性,并且证明,如果     

求解无限维非线性互补问题的光滑化牛顿法

研究一类无限维非线性互补问题的光滑化牛顿法.借助于非线性互补函数,将无限维非线性互补问题转化为一个非光滑算子方程.构造光滑算子逼近非光滑算子,在光滑逼近算子满足方向可微相容性的条件下,证明了光滑化牛顿法具有超线性收敛性.     

非角截断的Fokker-Plank-Boltzmann方程

在非角截断条件下研究了Fokker-Plank-Boltzmann方程的Cauchy问题,从而推广了DiPerna和Lions在角截断假设下的经典结果.主要利用了碰撞算子的结构的一些性质,以及对正规化碰撞算子的一种新的分解,并结合准椭圆算子的性质,得到了非截断条件下重正规化解的稳定性,进而得到了解的存在性.     

关于实效双曲算子的Cauchy问题解的C~∞-奇性分析

本文讨论了较大一类实效双曲算子的 Cauchy 问题的解在边界上重特征点附近的 C~∞-奇性传播情况.为此先找一个保持 Cauchy 问题形式不变的典则变换进行微局部化简,然后用拟基本解工具展开讨论.结果表明,尽管实效算子的Cauchy 问题的适定性与低阶项无关,但解的奇性在边界上重特征点附近出现分叉传播现象,且它紧密联系低阶项的性质.由本文结果就可研究所论算子的 Lax-Nirenberg 型的边界奇性反射问题.     

一类非线性退缩椭圆型方程Dirichlet问题无穷多个解的存在性

§1.引 言 设Ω R_+~n(n>1)是光滑有界区域,这里R_n~+={x=(x_1,x_2,…,x_n)∈R~n|x_1>0},且 Ω∩ R_+~n≠φ,我们研究边值问题此处m>0,因此A是二阶自伴退缩的椭圆算子,本文证明在m,p和h(x)的适当限制下,问题(1.1)存在无穷多个不同的解. 当m=0时,即A是二阶自伴一致椭圆算子,若h(x)≡0,则当     

一类非局部反应扩散方程组Cauchy问题的临界爆破指标

证明了一类来源于燃烧理论的非局部反应扩散方程组Cauchy问题解的局部存在性、唯一性及临界爆破指标的存在性。并证明临界爆破指标属于爆破情形。     

源于DNA的非线性波动方程组的Cauchy问题(英文)

本文首先证明源于DNA的非线性波动方程组的周期边值问题局部古典解的存在性和唯一性.其次通过周期边值问题序列证明这个方程组的Cauchy问题存在唯一的局部古典解.     

一类二阶线性椭圆型方程组Dirichlet问题按Hausdorff可解性

我们知道,对二阶椭圆组Dirichlet问题的可解性,特别是解的唯一性问题,许多作者都曾对不同类型的方程作过比较充分的讨论,但对破坏了唯一性,甚至是非Naether型可解的情况却讨论的很少。最初是,A.B.在[2]中例举了以下典型的二阶椭圆     

关于非主型偏微分方程的唯一性的一点注记

<正> 1.引言 关于主型偏微分方程的Cauchy问题的唯一性的研究,已经十分充分.事实上,经典的Cauchy-Kovalevsky定理断言,解析方程的非特征Cauchy问题具有唯一性;而Goursat定理则断言,特征Cauchy问题的解必是唯一的,如果特征支柱上只具有单特征的话(参看Hormander[1]第五章).但是,对非主型方程,情形就大不相同.1974年,Treves研     

关于线性偏微分方程的Cauchy问题的唯一性

<正> Cauchy问题的唯一性是偏微分方程的基本问题之一.经典的Cauchy-Kowalewski定理断言,解析方程或方程组的解析解是唯一的.1901年,Holmgren证明了,线性的解析方程或方程组的光滑解的唯一性.在取消关于系数的解析性的假设这个方向上的第一个结果是由Carleman在1939年给出的,他证明了两个自变量的相应结果,其中假设方程的主部的系数是实的,以及特征根是单重的,因而特征根的虚部如果不恒为零则总不为零.     

On the properties of ambiguity and irreversibility of dynamical maps of the initial data space of Cauchy problem

We investigate the Cauchy problems for evolutionary differential equations which possess the following properties: the solutions of considered problems admit the arising on the bounded time interval of singularities such that destroying of existence or uniqueness of solution and unbounded growth of norm of solution in Cauchy problem Banach space. The opportunity of continuation (probably of many-valued continuation) of the dynamical maps of the space of initial data by the procedure of passage to the limit for the sequences of approximating Cauchy problems is studied.     

On the lateral Cauchy problem

Considerable attention is currently begin devoted to Cauchy problems of mathemetical physics in view of their increasing importance for the solution of applied problems [M. Taylor, Partial Differential Equations, Springer, Berlin, 1966]. Uniqueness of the solution for these problems very important. In this paper we found a bound in the uniqueness theorem for the Cauchy problem.     

Picard and weakly Picard operators technique for nonlinear differential equations in Banach spaces

In this paper we study nonlocal Cauchy problems for differential equations in Banach spaces. Using Picard and weakly Picard operators technique and suitable Bielecki norms, some existence, uniqueness and data dependence results are obtained under some mild conditions. As an application, we also discuss a class of impulsive Cauchy problems by adapting the same methods.     

高阶拟线性拟双曲型方程组

在动物神经生物学问题与液体强迫振动等问题中,经常出现具有u_(11)-u_(xx1)项的线性或拟线性方程的各种问题。在[1]中对一般化的拟线性拟双曲型方程     

Cauchy problem for quasiparabolic factorized differential equations with variable domains of smoothed operators

We prove existence, uniqueness, and stability theorems for strong solutions of Cauchy problems for quasiparabolic factorized operator-differential equations with variable domains. For the first time, we derive a recursion formula for strong solutions of Cauchy problems, where recursion goes over the number of operator-differential factors in these equations. We prove the well-posed solvability (in the strong sense) for new mixed problems for partial differential equations with time-dependent coefficients in the boundary conditions.     

Generalized Cauchy type problems for nonlinear fractional differential equations with composite fractional derivative operator

This paper is devoted to proving the existence and uniqueness of solutions to Cauchy type problems for fractional differential equations with composite fractional derivative operator on a finite interval of the real axis in spaces of summable functions. An approach based on the equivalence of the nonlinear Cauchy type problem to a nonlinear Volterra integral equation of the second kind and applying a variant of the Banach’s fixed point theorem to prove uniqueness and existence of the solution is presented. The Cauchy type problems for integro-differential equations of Volterra type with composite fractional derivative operator, which contain the generalized Mittag-Leffler function in the kernel, are considered. Using the method of successive approximation, and the Laplace transform method, explicit solutions of the open problem proposed by Srivastava and Tomovski (2009) [11] are established in terms of the multinomial Mittag-Leffler function.     

Regularized Conservation Laws and Nonlinear Geometric Optics

 This paper is devoted to the study of Cauchy problems for regularized conservation laws in Colombeau algebras of generalized functions. The existence and uniqueness of generalized solutions to these Cauchy problems are obtained. Further, we develop a generalized variant of nonlinear geometric optics for the regularized problems. Consistency with the classical results is shown to hold for scalar conservation laws with bounded variation initial data in one space variable. Received 6 November 1996; in revised form 5 August 1997     

耗散Schrodinger-Poisson方程组的Cauchy问题

考虑耗散Schrodinger-Poisson方程组的Cauchy问题,对于吸引力场情形,证明了该问题整体强解的存在唯一性. 