On uniqueness and monotonicity of solutions of non-local reaction diffusion equation

This article deals with the uniqueness and the behavior of solutions of non-local reaction diffusion equations. Since these equations share many properties with the usual reaction diffusion model, such as a form of maximum principle and the translation invariance, uniqueness and monotone behavior for the solution, as in the usual case, are expected. I present an elementary proof of this monotone behavior. The proof essentially uses techniques based on the maximum principle and the sliding method.     

On Mayfield’s stability proof for the discretized transparent boundary condition for the parabolic equation

Mayfield’s numerical implementation of transparent boundary condition for the Schrödinger-type parabolic equations is revisited. An inaccuracy in the original proof of the conditional stability for the resulting scheme is pointed out. The highly unusual and impressive original result is reestablished and a new proof is presented. Some further remarks and estimates on the instability which occurs when the Mayfield condition is violated are given.     

Global Null Controllability of the 1-Dimensional Nonlinear Slow Diffusion Equation

The authors prove the global null controllability for the 1-dimensional nonlinear slow diffusion equation by using both a boundary and an internal control. They assume that the internal control is only time dependent. The proof relies on the return method in combination with some local controllability results for nondegenerate equations and rescaling techniques.     

第二类Painlevé方程解的亚纯性

Painlevé方程是六类重要的二阶代数微分方程,它们的发展和性质一直受到人们的关注.本文给出第二类Painlevé方程解的亚纯性一个严格的证明.     

关于一类HJ方程的Godunov通量的一点注记

本文对[1]中关于一类HJ方程的Godunov通量的命题作出改进,并给出新的证明.     

Infinite dimensional affine processes

The goal of this article is to investigate infinite dimensional affine diffusion processes on the canonical state space. This includes a derivation of the corresponding system of Riccati differential equations and an existence proof for such processes, which has been missing in the literature so far. For the existence proof, we will regard affine processes as solutions to infinite dimensional stochastic differential equations with values in Hilbert spaces. This requires a suitable version of the Yamada–Watanabe theorem, which we will provide in this paper. Several examples of infinite dimensional affine processes accompany our results.     

微分迭代方程的变换定理及其应用

本文给出了微分迭代方程的一个变换定理；作为应用，讨论了线性微分迭代方程ｘ（ｔ）＝ａｘ（ｔ） ｂｘ（ｘ（ｔ））在ａ＞ｂ＞０时单调解的存在性，纠正了文［４］讨论中的缺陷     

INTERACTION MORAWETZ ESTIMATE AND A SIMPLIFIED PROOF ON THE ENERGY SCATTERING FOR HARTREE EQUAITONS

In this article, we give a simple proof on the energy scattering for the Hartree equations using the interaction Morawetz estimate that was originally introduced in [5].     

一类函数方程的稳定性及微商配置解

本文对一类重要的函数方程建立解稳定性定理,提出微商配置解,证明了收敛性.     

弦割法与Muller法收敛阶的新证明方法

弦割法、Muller法与牛顿法一样,都是求解非线性方程的著名算法之一.然而在目前众多优秀的数值分析教材或论著中.关于弦割法和Muller法收敛阶的证明过程都是比较复杂的,无一例外的都是借助于差分方程的求解.本文对这两个算法的收敛阶给出了一种新的简单、直接的证明方法,达到了与牛顿法收敛阶证明方法的统一,同时还能够方便地求...     

Regular solutions to the coagulation equations with singular kernels

In this article, we prove the existence of solutions to the coagulation equation with singular kernels. We use weighted L¹‐spaces to deal with the singularities in order to obtain regular solutions. The Smoluchowski kernel is covered by our proof. The weak L¹ compactness methods are applied to suitably chosen approximating equations as a base of our proof. A more restrictive uniqueness result is also mentioned. Copyright © 2014 John Wiley & Sons, Ltd.     

一类Gronwall-Bellman型不等式的统一证明及其推广

借助一个极普通的比较定理,通过解普通的常微分方程,对一类Gronwall-Bellman型不等式及其推广进行了统一证明,并将其进一步推广.研究表明,这种方法极具实用性,利用这种方法我们可以构建几乎所有这种类型的不等式,从而为解决一类实际问题提供便捷的途径.     

Approximate controllability of a parabolic integrodifferential equation

In this paper, we obtain the approximate controllability of a parabolic integrodifferential equation with interior controls. The proof relies on the linear operator semigroup theory and the controllability theory for abstract equations. Copyright © 2013 John Wiley & Sons, Ltd.     

The Lazer-Solimini equation with state-dependent delay

Sufficient criteria are established for the existence of T-periodic solutions of a family of Lazer-Solimini equations with state-dependent delay. The method of proof relies on a combination of Leray-Schauder degree and a priori bounds.     

Blow-up rates for higher-order semilinear parabolic equations and systems and some Fujita-type theorems

In this paper, we derive blow-up rates for higher-order semilinear parabolic equations and systems. Our proof is by contradiction and uses a scaling argument. This procedure reduces the problems of blow-up rate to Fujita-type theorems. In addition, we also give some new Fujita-type theorems for higher-order semilinear parabolic equations and systems with the time variable on R. These results are not restricted to positive solutions.     

Existence and concentration of solutions for a class of biharmonic Kirchhoff equations with discontinuous nonlinearity

This paper concerns the existence and concentration of positive solutions for a class of biharmonic Kirchhoff equations with discontinuous nonlinearity. The proof relies on variational method, truncated methods, and nonsmooth critical points theory. Some related results are improved.     

Analysis and Applications of Extended Kantorovich–Krylov Method

In our prior work, the two-dimensional bending and in-plane mode shape functions of isotropic rectangular plates were solved based on the extended Kantorovich–Krylov method. These plate modes were then applied to sandwich plate analysis using the assumed modes method. Numerical results has shown these two-dimensional plate modes improved our sandwich plate analysis. However, the rigorous mathematical convergence proof of the extended Kantorovich–Krylov method is lacking. In this article, we provide a rigorous mathematical convergence proof of the extended Kantorovich–Krylov method using the example of rectangular plate bending vibration, in which the governing equation is a biharmonic equation. The predictions of natural frequency and mode shape functions based on the extended Kantorovich–Krylov method were calculated and the results were numerically validated by other analyses. A similar convergence proof can be applied to other types of partial differential equations (PDEs) that govern vibration problems in engineering applications. Based on these results, the extended Kantorovich–Krylov method was proven to be a powerful tooi for the boundary value problems of partial differential equations in the structural vibrations.     

Weak Dissipative Structure for Compressible Navier-Stokes Equations

This paper concerns the Cauchy problem for compressible Navier-Stokes equations.The weak dissipative structure is explored and a new proof for the classical solutions are shown to exist globally in time if the initial data is sufficiently small.     

Well-posedness of a Debye type system endowed with a full wave equation

We prove well-posedness for a transport-diffusion problem coupled with a wave equation for the potential. We assume that the initial data are small. A bilinear form in the spirit of Kato’s proof for the Navier–Stokes equations is used, coupled with suitable estimates in Chemin–Lerner spaces. In the one dimensional case, we get well-posedness for arbitrarily large initial data by using Gagliardo–Nirenberg inequalities.     

Exponential decay in time of solutions of the viscous quantum hydrodynamic equations

The long-time asymptotics of solutions of the viscous quantum hydrodynamic model in one space dimension is studied. This model consists of continuity equations for the particle density and the current density, coupled to the Poisson equation for the electrostatic potential. The equations are a dispersive and viscous regularization of the Euler equations. It is shown that the solutions converge exponentially fast to the (unique) thermal equilibrium state as the time tends to infinity. For the proof, we employ the entropy dissipation method, applied for the first time to a third-order differential equation.