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 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.

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.

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.

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.