UNIQUE CONTINUATION AND PERSISTENCE PROPERTIES OF SOLUTIONS OF THE 2-COMPONENT DEGASPERIS-PROCESI EQUATIONS

In this article, the unique continuation and persistence properties of solutions of the 2-component Degasperis-Procesi equations are discussed. It is shown that strong solutions of the 2-component Degasperis-Procesi equations, initially decaying exponentially together with its spacial derivative, must be identically equal to zero if they also decay exponentially at a later time.

ON UNIQUE CONTINUATION PROPERTIES FOR THE SUB-LAPLACIAN ON CARNOT GROUPS

In this article, authors begin with establishing representation formulas and properties for functions on Carnot groups. Then, some unique continuation results to solutions of sub-Laplace equations with potentials are proved.

ON DOUBLING PROPERTIES ANDTHE UNIQUE CONTINUATION AT THEBOUNDARY IN DINI DOMAINS

Let u be a solution to second order elliptic equations in Dini domains, a direct and elementary proof of the doubling property for u 2 over balls centered at points in the domain is presented. Moreover, under the zero Dirichlet boundary condition, the unique continuation at the boundary for Dini domains has been proved.

On Chinese Civilization Being the Only Continued Civilization in the Viewpoint of the Modern National History Construction

Chinese civilization is the only ancient civilization thriving to this day seems to be a common view in the historical academia and Chinese society as well. Culture is a way that human beings actively adapt to the surroundings and keep consistence with environmental change. All cultures created by human beings are changing, so is the Chinese culture. The physical and cultural evolution presented by the archaeological discoveries and related historical researches show that Chinese culture is multi-sourced and multicultural. This is the secret for the Chinese culture to retain its vitality. The reasons that Chinese civilization is the only civilization thriving to this day among all of ancient civilizations in the world include the ancestral worship, the identity to common ancestors, traditions of writing pedigrees and history books, the longhistory Chinese character system and the creative thinking and cultural methods of interpreting new thoughts by using traditional Chinese thinking and cultural structures. Especially in modern times, impacted by the nationalism trend, the national state narration built for meeting the requirement of the national state"historical memory"left us the strong impression of Chinese civilization being the only continued civilization and fostered the affective identification with the common history.

On the Unique Continuation Properties for Elliptic Operators with Singular Potentials

Let u be a solution to a second order elliptic equation with singular potentials belonging to Kato-Fefferman-Phong＇s class in Lipschitz domains. An elementary proof of the doubling property for u^2 over balls is presented, if the balls are contained in the domain or centered at some points near an open subset of the boundary on which the solution u vanishes continuously. Moreover, we prove the inner unique continuation theorems and the boundary unique continuation theorems for the elliptic equations, and we derive the Bp weight properties for the solution u near the boundary.

Unique Continuation on Quadratic Curves for Harmonic Functions

The unique continuation on quadratic curves for harmonic functions is dis-cussed in this paper.By using complex extension method,the conditional stability of unique continuation along quadratic curves for harmonic functions is illustrated.The nu-merical algorithm is provided based on collocation method and Tikhonov regularization.The stability estimates on parabolic and hyperbolic curves for harmonic functions are demonstrated by numerical examples respectively.

Optimal Three Cylinder Inequality at the Boundary for Solutions to Parabolic Equations and Unique Continuation Properties

Let Γ be a portion of a C^(1,α) boundary of an n-dimensional domain D. Letu be a solution to a second order parabolic equation in D x (-T, T) and assume that u = 0 on Γ x(-T, T), 0 ∈ Γ. We prove that u satisfies a three cylinder inequality near Γ x (—T, T). As aconsequence of the previous result we prove that if u(x,t) = O (|x|~k ) for every t ∈ (-T,T) andevery k ∈ N, then u is identically equal to zero.

Carleman estimates and unique continuation property for the anisotropic differential-operator equations

The unique continuation theorems for the anisotropic partial differential-operator equations with variable coefficients in Banach-valued L p -spaces are studied. To obtain the uniform maximal regularity and the Carleman type estimates for parameter depended differential-operator equations, the sufficient conditions are founded. By using these facts, the unique continuation properties are established. In the application part, the unique continuation properties and Carleman estimates for finite or infinite systems of quasielliptic partial differential equations are studied.

Unique Continuation Properties Operators with for Generalized Baouendi-Grushin Singular Weights

In this paper, properties of the spherical functions and Hardy-Sobolev inequalities of generalized Baouendi-Grushin vector fields are established, and then some unique continuation results for generalized Baouendi Grushin operators with singular weights are given.

Strong unique continuation property for a class of fourth order elliptic equations with strongly singular potentials

In this paper we prove the strong unique continuation property for a class of fourth order elliptic equations involving strongly singular potentials.Our argument is to establish some Hardy-Rellich type inequalities with boundary terms and introduce an Almgren’s type frequency function to show some doubling conditions for the solutions to the above-mentioned equations.

UNIQUE CONTINUATION ON A HYPERPLANE FOR WAVE EQUATION

One kind of unique continuation property for a wave equation is discussed. The authors show that, if one classical solution of the wave equation vanishes in an open set on a hyperplane, then it must vanish in a larger set on this hyperplane. The result can be viewed as a localized version of Robbiano’s result[9]. The approach involves the localized Fourier-Gauss transformation and unique continuation on a line in the Laplace equation.

CONTROL AND STABILIZATION OF THE KORTEWEG-DE VRIES EQUATION:RECENT PROGRESSES

The study of the control and stabilization of the KdV equation began with the work ofRussell and Zhang in late 1980s.Both exact control and stabilization problems have been intensivelystudied since then and significant progresses have been made due to many people's hard work andcontributions.In this article,the authors intend to give an overall review of the results obtained so farin the study but with an emphasis on its recent progresses.A list of open problems is also providedfor further investigation.

RAPID EXACT CONTROLLABILITY OF THE WAVE EQUATION BY CONTROLS DISTRIBUTED ON A TIME-VARIANT SUBDOMAIN

Consider the wave equation with distributed controls supported on a subdomain, calledcontrol subdomain, which is allowed to be variant in time. For any prescribed time duration,the authors work out a scheme for changing the control subdomain such that the wave equationis exactly controllable on this time duration, where the control subdomain at any time is allowedto have arbitrarily small measure and relatively simple shape.

Internal Controllability for Parabolic Systems Involving Analytic Non-local Terms

This paper deals with the problem of internal controllability of a system of heat equations posed on a bounded domain with Dirichlet boundary conditions and perturbed with analytic non-local coupling terms. Each component of the system may be controlled in a different subdomain. Assuming that the unperturbed system is controUable--a property that has been recently characterized in terms of a Kalman-like rank condition--the authors give a necessary and sufficient condition for the controllability of the coupled system under the form of a unique continuation property for the corresponding elliptic eigenvalue system. The proof relies on a compactness-uniqueness argument, which is quite unusual in the context of parabolic systems, previously developed for scalar parabolic equations. The general result is illustrated by two simple examples.

Carleman Estimates for the Schrdinger Equation and Applications to an Inverse Problem and an Observability Inequality

The authors prove Carleman estimates for spaces of negative orders, and use these estimates to problem of determining L^p-potentials. An L^2-1evel continuation results for the SchrSdinger equation are the Schrodinger equation in Sobolev prove the uniqueness in the inverse observability inequality and unique also obtained.

Refined Scattering and Hermitian Spectral Theory for Linear Higher-Order Schrōdinger Equations

The Cauchy problem for a linear 2mth-order Schrōdinger equation ut=-i（-△）^mu, in R^N×R＋,u｜t=0=u0;m≥1 is an integer,is studied, for initial data uo in the weighted space L^2ρ（R^N）,withρ^＊（x）=e｜x｜^a and a=2m/2m-1∈（1,2].The following five problems are studied： （I） A sharp asymptotic behaviour of solutions as t → ＋∞ is governed by a discrete spectrum and a countable set Ф of the eigenfunctions of the linear rescaled operator B=-i（-△）^m＋1/2my·↓△＋N/2mI,with the spectrum σ（B）={λβ=-｜β｜≥0}. （Ⅱ） Finite-time blow-up local structures of nodal sets of solutions as t → 0^- and a formation of ＂multiple zeros＂ are described by the eigenfunctions, being generalized Hermite polynomials, of the ＂adjoint＂ operator B=-i（-△）^m-1/2my·↓△,with the same spectrum σ（B^＊）=σ（B）.Applications of these spectral results also include： （Ⅲ） a unique continuation theorem, and （IV） boundary characteristic point regularity issues. Some applications are discussed for more general linear PDEs and for the nonlinear Schr6dinger equations in the focusing （＂＋＂） and defocusing （＂-＂） cases ut=-（-△）^mu±i｜u｜^p-1u,in R^N×R＋,where P〉1,as well as for： （V） the quasilinear Schr6dinger equation of a ＂porous medium type＂ ut=-（-△）^m（｜u｜^nu）,in R^N×R＋,where n〉0.For the latter one, the main idea towards countable families of nonlinear eigenfunctions is to perform a homotopic path n → 0^＋ and to use spectral theory of the pair {B,B^＊}.

A CARLEMAN ESTIMATE ON GROUPS OF HEISENBERG TYPE

A Pohozaev-Rellich type identity for the p-sub-Laplacian on groups of Heisenberg type, G, is given. A Carleman estimate for the sub-Laplacian on G is established and, as a consequence, a unique continuation result is proved.

Null Controllability of Some Reaction-Diffusion Systems with Only One Control Force in Moving Domains

In this article, the authors establish the local null controllability property for semilinear parabolic systems in a domain whose boundary moves in time by a single control force acting on a prescribed subdomain. The proof is based on Kakutani＇s fixed point theorem combined with observability estimates for the associated lineaxized system.

Some Theorems in the Theory of Microstretch Thermomagnetoelectroelasticity

The linear dynamic theory of microstretch thermomagnetoelectroelasticity is studied in this paper.First,a reciprocity relation which involves two processes at different instants is established to form the basis of a uniqueness result and a reciprocal theorem.The proof of the reciprocal theorem avoids both using the Laplace transform and incorporating the initial conditions into the equations of motion.The uniqueness theorem is derived with no definiteness assumption on the elastic constitutive coefficients.Then the continuous dependence theorem is discussed upon two external data systems.Finally,the variational principle of Hamilton type which fully characterizes the solution of the mixed boundary-initial-value problem（mixed problem） is obtained.These theorems lay a solid foundation for further theoretical and numerical studies on microstretch thermomagnetoelectroelastic materials.