A Fisher-Kolmogorov equation with finite speed of propagation

In this paper we study a Fisher-Kolmogorov type equation with a flux limited diffusion term and we prove the existence and uniqueness of finite speed moving fronts and the existence of some explicit solutions in a particular regime of the equation.     

PERIODIC SOLUTIONS TO A KIND OF RAYLEIGH EQUATION WITH A DEVIATING ARGUMENT AND IMPULSES

By employing the coincidence degree theory of Mawhin, we study the existence of periodic solutions to a kind of Rayleigh equation with a deviating argument and impulses. Some new results on the existence of periodic solutions to the equation are obtained.     

A three-dimensional analog of the Tricomi problem for a parabolic-hyperbolic equation

For a parabolic-hyperbolic equation, we study the three-dimensional analog of the Tricomi problem with a noncharacteritic plane on which the type of the equation changes. The uniqueness of the solution to the problem is proved by the method of a priori estimates, and the existence of a solution is reduced to the existence of a solution to a Volterra integral equation of the second kind.     

A nonlocal problem for a mixed-type equation whose order degenerates along the line of change of type

We consider a mixed-type equation whose order degenerates along the line of change of type. For this equation we study the unique solvability of a nonlocal problem with the Saigo operators in the boundary condition. We prove the uniqueness theorem under certain conditions (stated as inequalities) on known functions. To prove the existence of solution to the problem, we equivalently reduce it to a singular integral equation with the Cauchy kernel. We establish a condition ensuring the existence of a regularizer which reduces the obtained equation to a Fredholm equation of the second kind, whose unique solvability follows from that of the problem.     

A Radiation Condition for the 2-D Helmholtz Equation in Stratified Media

We study the 2-D Helmholtz equation in perturbed stratified media, allowing the existence of guided waves. Our assumptions on the perturbing and source terms are not too restrictive. We prove two results. Firstly, we introduce a Sommerfeld–Rellich radiation condition and prove the uniqueness of the solution for the studied equation. Then, by careful asymptotic estimates, we prove the existence of a bounded solution satisfying our radiation condition.     

A Nonlinear Degenerate Parabolic Equation

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A molecular method applied to a non-local PDE in stratified Lie groups

In this article we study a transport-diffusion equation in the framework of the stratified Lie groups. For this equation we will study the existence of the solutions, a maximum principle, a positivity principle and Hölder regularity.     

A multi-dimensional bistable nonlinear diffusion equation in a periodic medium

In this work we study the existence of wave solutions for a scalar reaction-diffusion equation of bistable type posed in a multi-dimensional periodic medium. Roughly speaking our result states that bistability ensures the existence of waves for both balanced and unbalanced reaction term. Here the term wave is used to describe either pulsating travelling wave or standing transition solution. As a special case we study a two-dimensional heterogeneous Allen–Cahn equation in both cases of slowly varying medium and rapidly oscillating medium. We prove that bistability occurs in these two situations and we conclude to the existence of waves connecting \(u = 0\) and \(u = 1\). Moreover in a rapidly oscillating medium we derive a sufficient condition that guarantees the existence of pulsating travelling waves with positive speed in each direction.     

一类泛函微分方程周期正解的存在性

本文研究了一类带参数的高维泛函微分程(1.1)的周期正解的存在性.利用一个著名的不动点指标定理,获得了该方程周期正解的存在性、多重性和不存在性.     

一类半线性椭圆型方程正整体解的存在性

本文研究了一个半线性椭圆型方程正整体解的存在性问题.利用上下解方法克服了非齐次项出现带来的困难,获得了正整体解存在的必要条件,具体地给出了在此条件下正整体解所处的领域.     

PERIODIC SOLUTIONS TO A HIGH-ORDER NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATION

In this paper,using the principle of a priori boundedness,we study the existence and uniqueness of periodic solutions to a high-order neutral functional differential equation.Some new results for the existence and uniqueness of the periodic solution are obtained.     

A Sobolev-like inequality for the Dirac operator

In this article, we prove a Sobolev-like inequality for the Dirac operator on closed compact Riemannian spin manifolds with a nearly optimal Sobolev constant. As an application, we give a criterion for the existence of solutions to a nonlinear equation with critical Sobolev exponent involving the Dirac operator. We finally specify a case where this equation can be solved.     

反射天线设计的一个数学理论

本文讨论反射天线面设计中出现的一个偏微分方程, 这是一个完全非线性Monge-Ampère型方程. 我们先给出一个广义解的定义, 然后介绍如何得到广义解的存在性、唯一性和正则性, 最后我们给出求数值解的一个方法.     

PERIODIC SOLUTIONS TO A KIND OF NEUTRAL DIFFERENTIAL SYSTEM:VIA (h,k)-DICHOTOMY

In this paper, based on the theory of (h, k)-Dichotomy of linear system and Kras-noselskii’s fixed point theorem, we study the existence of periodic solutions to a neutral differential equation. Some new sufficient conditions are obtained to guarantee the existence and uniqueness of T-periodic solution to the equation.     

A simplified quantum energy-transport model for semiconductors

The existence of global-in-time weak solutions to a quantum energy-transport model for semiconductors is proved. The equations are formally derived from the quantum hydrodynamic model in the large-time and small-velocity regime. They consist of a nonlinear parabolic fourth-order equation for the electron density, including temperature gradients; an elliptic nonlinear heat equation for the electron temperature; and the Poisson equation for the electric potential. The equations are solved in a bounded domain with periodic boundary conditions. The existence proof is based on an entropy-type estimate, exponential variable transformations, and a fixed-point argument. Furthermore, we discretize the equations by central finite differences and present some numerical simulations of a one-dimensional ballistic diode.     

A nonlinear age-dependent model with spatial diffusion

The main goal of this paper is the study of the existence and uniqueness of a positive solution for a nonlinear age-dependent equation with spatial diffusion. For that, we mainly use the properties of an eigenvalue problem related to the equation and the sub-supersolution method. We justify that this method works for this equation, in which there is a potential blow-up and a nonlocal initial condition.     

EXISTENCE OF MULTIPLE POSITIVE PERIODIC SOLUTIONS TO A CLASS OF INTEGRO-DIFFERENTIAL EQUATION

In this paper,by the Avery-Henderson fixed point theorem,we investigate the existence of multiple positive periodic solutions to a class of integro-differential equation. Some suficient conditions are obtained for the existence of multiple positive periodic solutions.     

Existence of solutions of non-autonomous second order functional differential equations with infinite delay

In this paper the existence of solutions of a non-autonomous abstract retarded functional differential equation of second order with infinite delay is considered. Assuming the existence of an evolution operator corresponding to the associate abstract Cauchy problem of second order, we establish the existence of mild solutions of the functional equation. Furthermore, we study the existence of classical solutions of the abstract Cauchy problem of second order and we apply these results to establish the existence of classical solutions of the functional equation. Finally, we apply our results to study the existence of solutions of the non-autonomous wave equation with delay.     

PROPERTIES OF POSITIVE SOLUTIONS FOR A NONLOCAL NONLINEAR DIFFUSION EQUATION WITH NONLOCAL NONLINEAR BOUNDARY CONDITION

This article deals with the global existence and blow-up of positive solution of a nonlinear diffusion equation with nonlocal source and nonlocal nonlinear boundary condition. We investigate the influence of the reaction terms, the weight functions and the nonlinear terms in the boundary conditions on global existence and blow up for this equation. Moreover, we establish blow-up rate estimates under some appropriate hypotheses.     

双曲空间中Laplace-Beltrami方程的一个带位移的边值问题

讨论了双曲空间中Laplace-Beltrami方程的一个带位移的边值问题.首先将双曲空间中的Laplace-Beltrami方程的解转化为Clifford分析中的超正则函数.然后给出了超正则函数的Plemelj公式并讨论了相关奇异积分算子的性质,最后利用积分方程的方法和压缩不动点原理证明了Laplace-Beltrami方程的一个带位移的边值问题的解的存在性和唯一性.