半线性高阶椭圆型方程非平凡解的存在性与不存在性

设(?)为 R~n 中的带光滑边界(?)的有界域。考察边值问题Δ~2u-aΔu bu=f(x,u),x∈(?),(1.1)或u=((?)u)/((?)v)=0,x∈(?) (1.2)u=Δu=0,x∈(?),(1.3)其中 a 和 b 为非负实数,((?)u)/((?)v)为沿(?)外法线的方向导数。当b=0,f(x,u)=cu~k,k为奇数且 c≤0时,文献[1]曾证明,方程(1.1)满足边值条件Δu|(?)=0的解满足极值原理;文献[2]则在对非线性项,f(x,u)加以某些限制的情况下,证明问题(1.1),(1.2)或(1.1),(1.3)存在非平凡解。本文的目的在于对上述问题作进一步讨论。在§2中,我们讨论了非平凡解存在性问题,代替[2]中的增长性条件     

MULTIPLE NONNEGATIVE SOLUTIONS OF FOURTH ORDER ORDINARY DIFFERENTIAL EQUATIONS

1IntroductionInthispaper)weconsidertheexistenceofmultiplenonnegativesolutionsoftheequationwi'thboundaryconditionandrealnumberA>0.Whenthefunctionf(y)inequation(1.l)isoftheformf(y)=y,theboundaryvalueproblem(1.1),(1.2)describesthedeformationsofanelasticbeambothofwhoseendssimplysupportedat0and1.In[1],MaRuyunstudiedtheequationHeprovedtheexistenceofatleastonepositivesolutioniffiseithersuperlinearorsublinear,i.e.OurpurposehereistogiveanexistenceresultformultiplenonnegativesolutionstoBVP(1.1),(1.2)…     

一类高维非自治系统的周期解

§1.引言在文献[1]中 Lasota-Opiul 对于非自治周期系统(?)=A(t,x)x b(t,x),(1.1)其中 A(t,x)是 n×n 连续矩阵,且 A(t ω,x)=A(t,x);b(t,x)是 n 维连续向量,且 b(t ω,x)=b(t,x).在“A(t,x)属于某一个 Banach 空间中的有界弱闭子集”的假设下,获得该系统周期解存在性定理.而这个假设条件不易验证,给定理的应用带来很大的不便.本文利用泛函分析的方法,借助于 Schauder 的不动点定理和矩阵测度的性质,对系统(1.1)的周期解的存在性进行了讨论.给出一个可以直接从系统(1.1)的右端函数性质来判别其周期解存在的定理.并且分别应用于系统(?)=A(t)x e(t),(1.2)     

沒有凸条件的一阶准线性方程的Cauchy問題的广义解的存在与唯一性

<正> 在凸条件f~〃(u)≠0下,一阶准线性议程的Cauchy问题■的間断解的存在与唯一性已經得到了相当充分的研究.某些力学問題,如象滤流理論,还要求研究f″(u)变号的情形. 的令人信服的分析指出,如下的条件似应足以确定Cauchy問題(1.1),(1.2)的間断解:u(t,x(t)-0)和u(t,x(t)+0)是函数     

具有源项的等熵气体动力学方程组全局解的存在性

本文我们证明了具有源项的等熵气体动力学方程组(1.2)全局解的存在性.为此,我们构造正则双曲方程组(1.1)去逼近非齐次等熵气体动力学方程组(1.2).首先,对每一个固定的逼近参数δ和一般化的P(ρ)条件,我们证明了带有界初始条件(1.4)的Cauchy问题(1.1)的全局熵解的存在性.其次,令∈=o(δ),我们得到了方程组(1.2)的形如η(ρ,u)=ρH(ρ,u)(与Chen和LeFloch(2003)相同)的弱熵对的H_(loc)~(-1)紧性的完整证明.最后,将Chen和LeFloch(2003)给出的关于P(ρ)的条件应用到定理1和定理2的结果中,我们得到了带有界初值(1.4)的Cauchy问题(1.2)的熵解的全局存在性.     

具有球对称结构的相对论欧拉方程组稳态解的存在性

对于等温气体,作者将考虑具有球对称结构的相对论欧拉方程组经典稳态解(定理1.1)和弱稳态解(定理1.2)的存在性.通过求解两个常微分方程以及比较马赫数M和1的关系,定理1.1证明了相对论欧拉方程组经典稳态解的存在性.在一个C~(2,α),α∈(0,1)稳态背景解的扰动下,定理1.2将证明高维球对称跨音速(双曲-椭圆)解的存在性.     

线性周期时滞微分方程解的结构

<正> 在[1]中我们讨论过下述线性时滞微分方程 x(t)=p(t)(x(t)-x(t-r)) (r>0为常数)(1.1)的解的结构,并得出结论: x(t)=c_ox_o(t)+x(t),(1.2)其中c_o为某个常数,x_o(t)为某个固定的无界解,而x(t)则为某个有界解. 本文讨论形式较(1.2)更一般的但系数为周期的时滞微分方程     

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     

PARALLEL STABILIZATION

1.IntroductionLetusconsideranevolutionsystemwhosestateisgivenbythesolutionofaPartialDifferentialEquation(PDE)whichiswritten(formallyforthetimebeing)asIn(1.1),(1.2),whichmaybealinearoranonlinearPDE,ydenotesthestate,andvdenotesthecontrol.TheoperatorA,w...     

关于具有初始挠度的圆薄板的跳跃问题

本文利用江福汝提出的“两变量法”[3][4]与正则摄动法研究了具有初始挠度的圆薄板的跳跃问题[本文中(1.1),(1.2)]。我们得到了这一问题的N阶一致有效渐近解[(1.66),(1.67)]。当初始挠度变为零时,该解变为圆薄板非线性弯曲问题的解[6];如果初始挠度较大而初始挠度与载荷强度符号相反时,载荷强度达到某一值时,将发生跳跃现象。     

On the Cauchy Problem of Evolution P-Laplacian Equations with Strongly Non-linear Sources When 1 < p < 2

We study the solvability of the Cauchy problem (1.1)–(1.2) for the largest possible class of initial values, for which (1.1)–(1.2) has a local solution. Moreover, we also study the critical case related to the initial value u ₀, for 1 < p < ∞. Project supported by the National Natural Science Foundation of China (19971070)     

Decay estimates for quasi-linear evolution equations

We consider global strong solutions of the quasi-linear evolution equations (1.1) and (1.2) below, corresponding to sufficiently small initial data, and prove some stability estimates, as t→+∞, that generalize the corresponding estimates in the linear case.     

Global existence and time decay estimate of solutions to the Keller–Segel system

We consider the chemotaxis‐Navier–Stokes system 1.1-1.4 (Keller–Segel system) in the whole space, which describes the motion of oxygen‐driven bacteria, eukaryotes, in a fluid. We proved the global existence and time decay estimate of solutions to the Cauchy problem 1.1-1.2 in with the small initial data. Moreover, when the fluid motion is described by the Stokes equations, we established the global weak solutions to 1.3-1.4 in with the potential function ? is small and the initial density n₀(x) has finite mass.     

The minimal genus problem in rational surfaces CP2 # n[`(CP2 )]CP^2 \# n\overline {CP^2 }

There are three key ingredients in the study of the minimal genus problem for rational surfaces : the generalized adjunction formula, the action of the orthogonal group of the Lorentz space and the geometric construction. In this paper, we prove the uniqueness of the standard form (see Definition 1.1 and Theorem 1.1) of a 2-dimensional homology class under the action of the subgroup of the Lorentz orthogonal group that is realized by the diffeomorphisms of . Using the geometric construction, we determine the minimal genera of some classes (see Theorem 1.2).     

Sur certains problèmes elliptiques asymétriques avec poids indéfinis

We prove the existence of a first nontrivial eigenvalue for the problem (1.2) below. As applications we study the solvability of (1.1) below as well as the Fučik spectrum with weights for the p-Laplacien.     

Lifespan of solutions of semilinear wave equations in two space dimensions

This paper studies the Cauchy problem for some doubly nonlinear degenerate parabolic equations (1.1) with initial data (1.2). Hölder continuous solutions, with explicit Hölder exponents uniformly in [0,T] * R^N for any given time T, are obtained by using the maximum principle.     

Exact controllability of semilinear abstract systems with application to waves and plates boundary control problems

This paper studies (global) exact controllability of abstract semilinear equations. Applications include boundary control problems for wave and plate equations on the explicitly identified spaces of exact controllability of the corresponding linear systems.Contents. 1. Motivating examples, corresponding results, literature. 1.1. Motivating examples and corresponding results. 1.2. Literature. 2. Abstract formulation. Statement of main result. Proof. 2.1. Abstract formulation. Exact controllability problem. 2.2. Assumptions and statement of main result. 2.3. Proof of Theorem 2.1. 3. Application: a semilinear wave equation with Dirichlet boundary control. Problem (1.1). 3.1. The case = 1 in Theorem 1.1 for problem (1.1). 3.2. The case = 0 in Theorem 1.1 for problem (1.1). 4. Application: a semilinear Euler—Bernoulli equation with boundary controls. Problem (1.14). 4.1. Verification of assumption (C.1): exact controllability of the linear system. 4.2. Abstract setting for problem (1.14). 4.3. Verification of assumptions (A.1)–(A.5). 4.4. Verification of assumption (C.2). 5. Proof of Theorem 1.2 and of Remark 1.2. Appendix A: Proof of Theorem 3.1. Appendix B: Proof of (4.9) and of (4.10b). References.Research partially supported by the National Science Foundation under Grant DMS-8902811 and by the Air Force Office of Scientific Research under Grant AFOSR-87-0321. The main results of this paper are announced in: Proceedings of the 28th Conference on Decision and Control, Tampa, Florida, December 1989, pp 2291–2294.     

F-空间中的一个基本定理及其在数值分析中的应用

本文针对F-空间中闭算子方程的一般逼近格式,研究其相容性、收敛性和稳定性之间的关系.所得的主要结果是：这种一般逼近格式在相容性条件下,其收敛性与稳定性是等价的.此定理可以看作是对Lax等价原理的推广,是求解第一类闭算子方程的一般逼近格式的基本定理.为得到这一主要结果,本文还给出了F-空间中的一条基本定理,众所周知的一致有界原理,闭图像定理和开映像定理是其简单推论.     

A Sturm–Liouville problem depending rationally on the eigenvalue parameter

We consider the Sturm–Liouville problem (1.1) and (1.2) with a potential depending rationally on the eigenvalue parameter. With these equations a λ ‐linear eigenvalue problem is associated in such a way that L₂‐solutions of (1.1), (1.2) correspond to eigenvectors of a linear operator. If the functions q and u are real and satisfy some additional conditions, the corresponding linear operator is a definitizable self‐adjoint operator in some Krein space. Moreover we consider the problem (1.1) and (1.3) on the positive half‐axis. Here we use results on the absense of positive eigenvalues for Sturm–Liouville operators to exclude critical points of the associated definitizable operator. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

一类含导数的非线性m-点边值问题的正解存在性

通过利用Krasnosel′skii不动点定理的扩充定理,对于一类含导数的非线性二阶m-点边值问题(1.1)+(1.2)u″(t)+f(t,u(t),u′(t))=0,0相似文献