On a Class of Elliptic-Parabolic Equations on Unbounded Intervals

We study a class of degenerate elliptic second order differential operators acting on some polynomial weighted function spaces on [0,+[. We show that these operators are the generators of C ₀-semigroups of positive operators which, in turn, are the transition semigroups associated with right-continuous normal Markov processes with state space [0,+]. Approximation and qualitative properties of both the semigroups and the Markov processes are investigated as well. Most of the results of the paper depend on a representation of the semigroups we give in terms of powers of particular positive operators of discrete type we introduced and studied in a previous paper.     

Formulation and well-posedness of the Cauchy problem for a diffusion equation with discontinuous degenerating coefficients

The choice of a differential diffusion operator with discontinuous coefficients that corresponds to a finite flow velocity and a finite concentration is substantiated. For the equation with a uniformly elliptic operator and a nonzero diffusion coefficient, conditions are established for the existence and uniqueness of a solution to the corresponding Cauchy problem. For the diffusion equation with degeneration on a half-line, it is proved that the Cauchy problem with an arbitrary initial condition has a unique solution if and only if there is no flux from the degeneration domain to the ellipticity domain of the operator. Under this condition, a sequence of solutions to regularized problems is proved to converge uniformly to the solution of the degenerate problem in L ₁(R) on each interval.     

Positivity and Stability for a Population Equation with Diffusion on 1

Using positivity, abstract extrapolation spaces, and a generalization of the Jörgens–Vidav-Voigt perturbation theorem for the essential spectral radius of strongly continuous semigroups given in [19], we study the asymptotic behavior of a linear age-dependent population problem with spatial diffusion where the birth and the death rates depend on the spatial variable.     

抛物型方程的解熄灭（extinction）的充要条件

本文给出了某些抛物型方程初边值问题解熄灭的充分和必要的条件。     

线性Schrdinger方程的初边值问题(英文)

本文证明了非线性 Ｓｃｈｒｏｄｉｎｇｅｒ方程的初边值问题的 Ｈｓ－解（１＜ ｓ ≤ ２）的存在唯一性结果．     

具有Boltzmann阻尼的Petrovsky系统的稳定性

研究具有Boltzmann阻尼的Petrovsky系统的稳定性.首先在合适的假设下,应用线性算子半群理论证明了系统的适定性,进而运用Hilbert空间线性算子半群指数稳定的频域结果证明了具有Boltzmann阻尼Petrovsky系统的指数稳定性.     

一类拟线性退化抛物型方程的初边值问题

本文讨论下面方程的初边值问题。光滑有界区域，且ｐ＞２．本文用正则化方法证明了当ｐ＞α＋２时广义解的全局存在性，对ｐ＜α＋２的情形，证明当初值ｕ＿ｏ（ｘ）属于一稳定集Ｗ时广义解的全局存在性，而当ｕ＿ｏ（ｘ）充分大时广义解只局部存在并在有限时间内爆破。最后用共轭算子法得到了广义解的唯一性定理。     

Generalized Kantorovich Operators on Bauer Simplices and Their Limit Semigroups

In this paper, we prove an asymptotic formula for generalized Kantorovich operators associated with the canonical Markov projection on a given Bauer simplex K. That formula involves an operator acting on the subalgebra of all products of a?ne functions on K. Moreover, we prove that such an operator is closable and its closure is the generator of a Markov semigroup which, in turn, may be represented in terms of iterates of the above-mentioned generalized Kantorovich operators.     

A RIEMANN-HILBERT APPROACH TO THE INITIAL-BOUNDARY PROBLEM FOR DERIVATIVE NONLINEAR SCHRODINGER EQUATION

We use the Fokas method to analyze the derivative nonlinear Schrodinger （DNLS） equation iqt （x, t） = -qxx （x, t）＋（rq^2）x on the interval [0, L]. Assuming that the solution q（x, t） exists, we show that it can be represented in terms of the solution of a matrix Riemann- Hilbert problem formulated in the plane of the complex spectral parameter ξ. This problem has explicit （x, t） dependence, and it has jumps across {ξ∈C｜Imξ^4 = 0}. The relevant jump matrices are explicitely given in terms of the spectral functions {a（ξ）, b（ξ）}, {A（ξ）, B（ξ）}, and {A（ξ）, B（ξ）}, which in turn are defined in terms of the initial data q0（x） = q（x, 0）, the bound- ary data g0（t）= q（0, t）, g1（t） = qx（0, t）, and another boundary values f0（t） = q（L, t）, f1（t） = qx（L, t）. The spectral functions are not independent, but related by a compatibility condition, the so-called global relation.     

Some classes of inverse evolution problems for parabolic equations

We consider the problem of simultaneously determining coefficients of a second order nonlinear parabolic equation and a solution to this equation. The unknown coefficients occur in the main part and in the nonlinear summand as well. The overdetermination conditions are conditions of the Dirichlet type on a family of planes of arbitrary dimension. It is demonstrated that the problem in question is solvable locally in time in Hölder spaces. When the unknown functions enter the right-hand side and the equation is linear, the theorem of global unique existence (in time) is established.     

MULTIPLE POSITIVE SOLUTIONS FOR SINGULAR BOUNDARY VALUE PROBLEMS WITH p-LAPLACIAN

This paper is concerned with the existence of positive solutions to singular boundary value problems of functional differential equations with p-Laplacian. Under the suitable conditions,the existence of three positive solutions is esta- blished by using an abstract fixed point theorem.     

n阶变系数线性差分方程的解

本文利用变数算符￣［２］以及给出变数算符和移动算符的乘积关系，并定义变系数移动算符幂级数间的乘积且证明其在Ｍｉｋｕｉｕｓｋｉ收敛意义下是正确的；另外，把一般的ｎ阶变系数线性差分方程转化为一个恰当的算符方程组，从而获得一般ｎ阶变系数线性差分方程的解。     

ON THE TRANSPORT EQUATION WITH INTEGRAL BOUNDARY CONDITIONS AND ITS CONSERVAIVE LAW

The well-posedness for the time-dependent neutron transport equationwith integral boundary conditions is established in L~1 space. Some spectral propertiesof the transport operator are discussed, the dominant eigenvalue is proved existing,and furthermore, the conservative law of migrating particle numbers is established.     

带奇异项的半线性抛物方程全局解的渐近分析

本文考虑一类带奇异项的半线性抛物方程的初边值问题，利用标准的正则摄动方法和抛物方程的极值原理，对任意给定的正整数Ｎ、得到上述问题的全局古典解的Ｎ＋１阶一致近似计算公式。