A novel parameter-uniform finite difference scheme on a Shishkin-type mesh for a singularly perturbed Volterra integro-differential equation is studied. The problem is discretized by the variable two-step backward differentiation formula (BDF2) for the first-order derivative term and the trapezoidal formula for the integral term. The stability of the proposed numerical method is carried out. It is shown from the convergence analysis that our presented method is almost second-order uniformly convergent with respect to the perturbation parameter ε in the discrete maximum norm. Numerical results are given to support our theoretical result.
In this paper, we study Hessian equations with the prescribed contact angle boundary value or oblique derivative boundary value and finally derive the a priori global gradient estimate for the admissible solutions.
The global regularity problem concerning the inviscid Boussinesq equations remains an open problem. In an attempt to understand this problem, we examine the damped Boussinesq equations and study how damping affects the regularity of solutions. In this paper, we consider the global existence to the damped Boussinesq equations with a class of large initial data, whose $B^{s}_{p,r}$ or $\dot{B}^{s}_{p,r}$ norms can be arbitrarily large. The idea is splitting the linear Boussinesq equations from the damped Boussinesq equations, the exponentially decaying solution of the former equations together with the structure of the Boussinesq equations help us to obtain the global smooth solutions.
In the present paper, we prove the existence, non-existence and multiplicity of positive normalized solutions (λc, uc) ∈ ℝ × H1 (ℝN) to the general Kirchhoff problem -M(∫_ℝ^N|∇ u|^2 dx)Δ u +λ u=g(u) in ℝ^N, u∈ H^1(ℝ^N),N≥ 1, satisfying the normalization constraint ∫_ℝ^Nu^2 dx=c , where M ∈ C([0, ∞)) is a given function satisfying some suitable assumptions. Our argument is not by the classical variational method, but by a global branch approach developed by Jeanjean et al. [J Math Pures Appl, 2024, 183: 44–75] and a direct correspondence, so we can handle in a unified way the nonlinearities g(s), which are either mass subcritical, mass critical or mass supercritical.
This research studies the inverse boundary value problem for fractional elliptic equation of Tricomi–Gellerstedt–Keldysh type and obtains a condition stability result. To recover the continuous dependence of the solution on the measurement data, a generalized Tikhonov regularization method based on ill-posedness analysis is constructed. Under the a priori and a posterior selection rules for the regularization parameter, corresponding Hölder type convergence results are obtained. On this basis, this thesis verifies the simulation effect of the generalized Tikhonov method through numerical examples. The examples show that the method performs well in dealing with the problem under consideration.
The symmetric realizations of the product of two higher-order quasi-differential expressions in Hilbert space are investigated. By means of the construction theory of symmetric operators, we characterize symmetric domains determined by two-point boundary conditions for product of two symmetric differential expressions with regular or limit-circle singular endpoints. The presented result contains the characterization of self-adjoint domains as a special case. Several examples of singular symmetric product operators are given.
In this paper, we consider the existence of multiple solutions of the following critical nonlocal elliptic equations with magnetic field: {(-i del - A(x))(2)u = lambda|u|(p-2)u + (integral(Omega) |u(y)(2 alpha*)/|x - y|(alpha) dy) |u|(2 alpha*-2u) in Omega, u = 0 partial derivative Omega, (1) where i is imaginary unit, N >= 4, 2(alpha)* = 2N-alpha/N-2 with 0 < alpha < 4, lambda > 0 and 2 <= p < 2* = 2N/N-2 . Suppose the magnetic vector potential A(x) = (A(1)(x), A(2)(x), ..., A(N)(x)) is real and local Holder continuous, We show by the Ljusternik-Schnirelman theory that (1) has at least cat(Omega)(Omega) nontrivial solutions for lambda small.
In this article,we introduce the concepts of Weyl-mean equicontinuity and Weyl-mean sensitivity of a random dynamical system associated to an infinite countable discrete amenable group action.We obtain the dichotomy result to Weyl-mean equicontinuity and Weyl-mean sensitivity of a random dynamical system when the corresponding skew product transformation is minimal.
This paper studies the global regularity problem for the 2D micropolar Rayleigh–Bénard convection system with velocity zero dissipation, micro‐rotation velocity Laplace dissipation, and temperature critical diffusion. By introducing a combined quantity and using the technique of Littlewood–Paley decomposition, we establish the global regularity result of solutions to this system.
In this paper, we propose an algorithm combining Bregman alternating minimization algorithm with two-step inertial force for solving a minimization problem composed of two nonsmooth functions with a smooth one in the absence of convexity. For solving nonconvex and nonsmooth problems, we give an abstract convergence theorem for general descent methods satisfying a sufficient decrease assumption, and allowing a relative error tolerance. Our result holds under the assumption that the objective function satisfies the Kurdyka–Łojasiewicz inequality. The proposed algorithm is shown to satisfy the requirements of our abstract convergence theorem. The convergence is obtained provided an appropriate regularization of the objective function satisfies the Kurdyka–Łojasiewicz inequality. Finally, numerical results are reported to show the effectiveness of the proposed algorithm.
In this paper,we study some characterizations of Hankel operators on vector-valued ex-ponential type weights Bergman spacesA2φ(H)induced by operator-valued function symbols and co-analytic operator-valued function symbols.Main results including the boundedness and compactness of Hankel operators.
In this paper we study exponential tractability of multivariate approximation problem for weighted Korobov spaces in the worst case setting. The considered algorithms are constructive and use the class Λstd of only function evaluations. We give matching necessary and sufficient conditions for notions of EC-quasi-polynomial tractability and EC-uniform weak tractability in terms of two weight parameters of the problem.
We consider convergence sets of formal power series f(z,t)=∞∑n=0 fn(z)tn,where fn(z)are holomorphic functions on a domain Ω in C.A subset E of Ω is said to be a convergence set in Ω if there is a series f(z,t)such that E is exactly the set of points z for which f(z,t)converges as a power series in a single variable t in some neighborhood of the origin.A σ-convex set is defined to be the union of a countable collection of polynomially convex compact subsets.We prove that a subset of C is a convergence set if and only if it is σ-convex.
In this article, we investigate a class of time feedback optimal control systems governed by Caputo fractional semilinear differential equations in Banach space. At first, we discuss the existence and uniqueness of the mild solution for the equations by using fixed point theorem. Secondly, we show that the admissible trajectories set is non-empty involving the compactness of semigroup T(t) (t > 0) with the help of the Cesari property and the Fillippove theorem. Moreover, we also give an existence result for time feedback optimal control. In the end, an example is given to illustrate our main results.
This paper is concerned with the existence of positive solutions to the fourth-order boundary value problem u(4)(x)=f(x,u(x),u″(x)) on the interval [0, 1] with the boundary condition u(0)=u(1)=u″(0)=u″(1)=0, which models a statically bending elastic beam whose two ends are simply supported. Without assuming that the nonlinearity f(x, u, v) is nonnegative, an existence result of positive solutions is obtained under the inequality conditions that |(u, v)| is small or large enough. The discussion is based on the method of lower and upper solutions.
In this paper, we study Finsler warped product metrics. We obtain the differential equations that characterize Landsberg Finsler warped product metrics. By solving these equations, we obtain the expression of these metrics. Furthermore, we construct a class of almost regular Finsler warped product metrics F with the following properties: (1) F is a Landsberg metric; (2) F is not a Berwald metric; (3) F has zero flag curvature (or Ricci curvature).
利用概率变换思路,基于VG分布提出了VG扭曲算子.在VG模型中,证明了按VG扭曲算子得到的期权价格和在均值修正鞅测度下的期权价格一致.数值计算结果表明,按VG扭曲算子得到的期权价格比较准确.
该文建立了 Fock型空间上单边加权移位算子的Schödinger测不准关系,并给出了等号成立时的显式表达,进而推广了文献[4]中建立的Fock空间上Heisenberg型测不准关系并克服了文献[16]中的困难.该文进一步将结果推广到多个算子情形,还得到了单边加权移位算子的一个非自伴形式的测不准不等式.
该文致力于研究带部分调和势的非齐次非线性Schrödinger方程的Cauchy问题.该方程是玻色-爱因斯坦凝聚中的一个重要模型.结合非线性椭圆方程基态解的变分特征及质量和能量守恒,首先得到了该问题整体解的存在性,并利用尺度变换技巧证明了该方程在一些特殊初值情形下存在爆破解.其次讨论了爆破解的L2集中现象.最后利用与上述基态解相关的变分结论研究了 L2最小质量爆破解的动力学性质,即具有最小质量的爆破解的极限profile、精细质量集中和爆破速率.该文将Zhang[35]的全局存在性和爆破结果推广到带非齐次非线性项的情形,并将Pan和Zhang[24]的部分结果改进到空间维数N＞2且非线性项为非齐次的情形.
该文研究了R上几类权函数为加倍权的条件.首先给出了R上单调权函数为加倍权的充要条件;其次刻画了R上分段单调权函数为加倍权的条件;最后讨论了 R上分段加倍权函数为加倍权的条件.
