Nonexistence and existence of positive solutions for the Kirchhoff type equation

In this paper, we study the nonexistence and existence of positive solutions for the Kirchhoff type equation with nonlinearity having prescribed asymptotic behavior. Because of the structure of our equation, our nonexistence results have some special properties, and our conditions for positive solutions are also different from the existing results.     

Dynamical interface transition in ramified media with diffusion

We consider interaction problems in ramified spaces of a rather general type with a distinguished interface transition in form of a dynamical Kirchhoff condition. These conditions stem from applications and modeling in biology and physics, see for instance [8,21]. In the present paper our principal concern is to derive existence results in the linear and semilinear case and some qualitative principles similar to the classical ones for initial boundary value problems on domains. Moreover, the influence of the dynamical interface condition on the solution is studied in the autonomous reaction-diffusin case.     

有限变形极性弹性介质的各型动力学方程组*

本文从Dluzewski提出的以欧拉角为角坐标的建议和推导出的Cauchs型动力学方程组出发,又引进若干有关变形几何学和动力学的新定义并推导出有限变形极性弹性介质的Boussinesq型、Kirchhoff型、Signorini型和Новожилов型动力学方程组．     

Kirchhoff方程单峰解的局部唯一性

该文主要证明了以下非线Kirchhoff问题的单峰解的局部唯一性-(∈^2a+∈b∫R^3|▽u|^2dx)△u+u=K(x)|u|^p-1u,u> 0,x∈R^3,其中∈>0任意小,a,b> 0,1相似文献   

ASYMPTOTIC BEHAVIOR OF WAVE EQUATION OF KIRCHHOFF TYPE WITH STRONG DAMPING

The paper deals with the strongly damped nonlinear wave equation of Kirchhoff type. The existence of a global attractor is proven by using the decomposition, and moreover, the structure of the global attractor is established. Our results improve the previous results.     

On spectral asymptotics and bifurcation for some elliptic equations of Kirchhoff-type with odd superlinear term

In this paper, from estimating the eigenvalues for Kirchhoff elliptic equations, we obtain spectral asymptotics and bifurcation concerning the eigenvalues of some related elliptic linear problem.     

Solutions of the autonomous Kirchhoff type equations in RN

Recently, the existence, the multiplicity and the concentration of solutions for the Kirchhoff type equations or systems have been extensively established by a number of authors. Such problems contain a nonlocal term which implies that the equation or system is no longer a pointwise identity. Thus, some researchers think that the nonlocal phenomenon will cause some mathematical difficulties. In this note, however, we will give a simple transformation so that the solutions of the autonomous Kirchhoff type equation or system are easily obtained by using the known solutions of the corresponding local equation or system. In particular, some qualitative properties of solutions for the local problems are also inherited.     

Exponential stabilization of hyperbolic systems with nonlinear,unbounded perturbations—riccati operator approach

Abstract differential equations with nonlinear unstructured perturbations represented by unbounded nonlinear operators are considered. It is shown that such system can be uniformly locally stabilized by the feedback operator (also unbounded) which is constructed via the solution of an appropriate Riccati Equation. Abstract results are applied to the model of a Kirchhoff plate with nonlinear unstruc¬tured boundary perturbations. In this case, it is proved that the energy of the solutions with boundary (moment) feedback based on Riccati operator decays uniformly (locally) to zero     

Sharp trace estimates of solutions to Kirchhoff and Euler-Bernoulli equations

This paper gives sharp estimates of second-order traces of solutions to mixed problems for Kirchhoff equations and Euler-Bernoulli equations with second- and third-order nonhomogeneous boundary conditions. The approach uses pseudodifferential analysis. These sharp trace regularity results have, among others, important implications in corresponding exact controllability/uniform stabilization theories; they allow the removal of geometrical conditions made in prior literature.This research was partially supported by the National Science Foundation under Grant DMS-89-02811.     

One-bump positive solutions for the Kirchhoff equations with perturbation in

In this paper, we study the Kirchhoff equations with perturbation in . Applying the finite reduction method, we prove that the equation has one-bump positive solutions under some suitable conditions that are given in Section 1.     

Partial regularity for harmonic maps of evolution into spheres

We consider the global Cauchy problem for generalized Kirchhoff equations with small non-linear terms or small data. We solve this problem in the space of functions which are twice differentiable with respect to time coordinate and uniformly analytic with respect to other coordinates. We determine, in two different situations, estimates of lifespan of solutions for some problems with perturbations and we give stability result of the solution for small perturbations.     

Stabilization by memory effects: Kirchhoff plate versus Euler–Bernoulli plate

In this paper we study the asymptotic behavior of solutions of a dissipative coupled system where we have interactions between a Kirchhoff plate and an Euler–Bernoulli plate. The dissipative mechanism is given by memory terms that act either collaboratively (in both equations) or unilaterally (in only one equation). We show that the solutions of this system decay to zero sometimes exponentially and other times polynomially. We found explicit decay rates that depend on the fractional exponents of the memory in each of the following cases: when the memory only acts in the Kirchhoff equation, or only in the Euler–Bernoulli equation, or in both. We also show that all decay rates found are the best. The results obtained are surprising for the following facts: in the collaborative case, the best decay rates of the system are given by the worst decay rates of the uncoupled equations, and in the unilateral case, we conclude that the memory effects in the Euler–Bernoulli equation dissipate the system more slowly than memory effects in the Kirchhoff equation.     

Variational approaches to impulsive elastic beam equations of Kirchhoff type

In this paper, we study the existence of multiple solutions for impulsive fourth-order differential equations of Kirchhoff type. Using a variational method and some critical points theorems, we obtain some new criteria for guaranteeing that impulsive fourth-order differential equations of Kirchhoff type have three and infinitely many solutions. Some recent results are extended and improved. Some examples are presented to demonstrate the applications of our main results.     

Equivalent Conditions for the Solvability of Nonstandard and Singular LQ-Problems with Application to Parabolic Equations

We consider an optimal control problem with indefinite cost for an abstract model, which covers, in particular, parabolic systems in a general bounded domain. Necessary and sufficient conditions are given for the synthesis of the optimal control, which is given in terms of the Riccati operator arising from a nonstandard Riccati equation. The theory extends also a finite-dimensional frequency theorem to the infinite-dimensional setting. Applications include the heat equation with Dirichlet and Neumann controls, as well as the strongly damped Euler–Bernoulli and Kirchhoff equations with the control in various boundary conditions.     

正交曲线坐标系下极性连续统的动力学方程组

应用非完整系物理标架给出正交曲线坐标系下Bousinesq型、Kirchhof型、Signorini型和Novozhilov型有限变形极性弹性介质动力学方程组的具体形式.     

Finite element analysis of parabolic integro-differential equations of Kirchhoff type

The aim of this paper is to study parabolic integro-differential equations of Kirchhoff type. We prove the existence and uniqueness of the solution for this problem via Galerkin method. Semidiscrete formulation for this problem is presented using conforming finite element method. As a consequence of the Ritz–Volterra projection, we derive error estimates for both semidiscrete solution and its time derivative. To find the numerical solution of this class of equations, we develop two different types of numerical schemes, which are based on backward Euler–Galerkin method and Crank–Nicolson–Galerkin method. A priori bounds and convergence estimates in spatial as well as temporal direction of the proposed schemes are established. Finally, we conclude this work by implementing some numerical experiments to confirm our theoretical results.     

极性连续统的增率型运动方程和边界条件

推导出了各种偶应力张量间和它们的变率间的关系,并建立起增率型角动量方程及其相应的边界条件。于是,把这些结果和匡震邦在“非线性连续介质力学基础”中给出的经典连续统力学的相应结果组合起来即得Cauchy形式和Piola形式以及Kirchhoff形式的极性连续统的增率型运动方程和边界条件。     

Remarks on non homogeneous elliptic Kirchhoff equations

Extending the results in (Ambrosetti and Arcoya, Positive solutions of elliptic Kirchhoff equations (to appear)), the existence of solutions for some classes of nonlinear elliptic, possibly degenerate, non homogeneous Kirchhoff equations are proved. We handle smooth nonlinearities as well as the case in which the nonlinearity has an upward jump discontinuity.     

EQUIVALENCE BETWEEN EXACT INTERNAL CONTROLLABILITY OF THE KIRCHHOFF PLATE-LIKE EQUATION AND THEWAVE EQUATION

1.IntroductionConsiderthefollowingillitial-boundaryvalueproblemLetD(A)={vEHI(fl)ItvEL'(fl)},V~Ha(~~),H~L'(fl).Thenthefiniteenergystatespacesofthesystems(1.1),(1.2)are,respectively,uk~D(A)xV,U.~VxH.ConsiderthegeneralizedfunctionspaceH-1~V'DHDV.ForbEC'(fl)anduEH-',wedefinebaEH-1by(bu)(()~u(bo,V(6HJ(n).(1.3)WehaveusedtheSobolevspaces.Wereferthereadersto[1]fortheknowledgeofthosespaces.Definition1.1.Thesystem(1.1)issaidtobeexactlycontrollableinukbyH-1-controlsifforevery(yo,yi)6uk,th…     

Global solutions to kirchhoff equations with non-local nonlinearities defined by a potential

In this paper we investigate the global solvability for small initial data of Kirchhoff type equations is a suitable potential. Besides, we will give a partial characterization of the asymptotic behavior of the solutions as