uniqueness

摘要： This article is concerned with a strongly coupled elliptic system modeling the steady state of two or more populations that compete in some regions. We prove the uniqueness of the limiting configuration as the competing rate tends to infinity, under suitable conditions. The proof relies on properties of limiting solution and Maximum principle.

摘要： We study a nonlinear equation in the half-space with a Hardy potential,specifically,−Δ_(p)u=λu^(p−1)x_(1)^(p)−x_(1)^(θ)f(u)in T,where Δp stands for the p-Laplacian operator defined by Δ_(p)u=div(∣Δu∣^(p−2)Δu),p>1,θ>−p,and T is a half-space{x_(1)>0}.When λ>Θ(where Θ is the Hardy constant),we show that under suitable conditions on f andθ,the equation has a unique positive solution.Moreover,the exact behavior of the unique positive solution as x_(1)→0^(+),and the symmetric property of the positive solution are obtained.

摘要： In this paper,we extend our previous result from Lévy(2016).We prove that transport equations with rough coefficients do possess a uniqueness property,even in the presence of viscosity.Our method relies strongly on duality and bears a strong resemblance to the well-known DiPerna-Lions theory first developed by DiPerna and Lions(1989).This uniqueness result allows us to reprove the celebrated theorem of Serrin(1962)in a novel way.As a byproduct of the techniques,we derive an L^(1) bound for the vorticity in terms of a critical Lebesgue norm of the velocity field.We also show that the zero solution is unique for the 2D Euler equations on the torus under a mild integrability assumption.

摘要： A huge amount of sensitive personal data is being collected by various online health monitoring applications.Although the data is anonymous,the personal trajectories(e.g.,the chronological access records of small cells)could become the anchor of linkage attacks to re-identify the users.Focusing on trajectory privacy in online health monitoring,we propose the User Trajectory Model(UTM),a generic trajectory re-identification risk predicting model to reveal the underlying relationship between trajectory uniqueness and aggregated data(e.g.,number of individuals covered by each small cell),and using the parameter combination of aggregated data to further mathematically derive the statistical characteristics of uniqueness(i.e.,the expectation and the variance).Eventually,exhaustive simulations validate the effectiveness of the UTM in privacy risk evaluation,confirm our theoretical deductions and present counter-intuitive insights.

摘要： The author has retracted this article because there is a flaw in the proof of Theorem 2.4 which invalidates the subsequent inequalities.

摘要： In this paper, we apply the Adomian decomposition method (ADM) for solving nonlinear system of fractional differential equations (FDEs). The existence and uniqueness of the solution are proved. The convergence of the series solution and the error analysis are discussed. Some applications are solved such as fractional-order rabies model.

摘要： In this paper,we study the uniqueness of entire functions and prove the following theorem.Let f be a transcendental entire function of finite order.Then there exists at most one positive integer k,such that f(z)△^(k)_(c)f(z)-R(z)has finitely many zeros,where R(z)is a non-vanishing rational function and c is a nonzero complex number.Our result is an improvement of the theorem given by Andasmas and Latreuch[1].

摘要： In recent years, a vast amount of work has been done on initial value problems for important nonlinear evolution equations like the nonlinear Schrödinger equation (NLS) and the Korteweg-de Vries equation (KdV). No comparable attention has been given to mixed initial-boundary value problems for these equations, i.e. forced nonlinear systems. But in many cases of physical interest, the mathematical model leads precisely to the forced problems. For example, the launching of solitary waves in a shallow water channel, the excitation of ion-acoustic solitons in a double plasma machine, etc. In this article, we present the PDE (Partial Differential Equation) method to study the following iut = uxx - g|u|pu, g ∈ R, p > 3, x?∈ Ω = [0,L], 0 ≤?t?u (x,0) = u0 (x) ∈?H2 (Ω) and Robin inhomogeneous boundary condition ux (0,t) + αu (0,t) = R1(t), t ≥ 0 and ux (L,t) + αu (L,t) = R2 (t), t ≥ 0 (here?α?is a real number). The equation is posed in a semi-infinite strip on a finite domain Ω. Such problems are called forced problems and have many applications in other fields like physics and chemistry. The main tool of PDE method is semi-group theory. We are able to prove local existence and uniqueness theorem for the nonlinear Schrödinger equation under initial condition and Robin inhomogeneous boundary condition.