Forward dynamical problem for the Timoshenko beam

The initial boundary value problem

An existence principle for solutions to a singular boundary-value problems

The singular boundary-value problem

$ \left\{ {\begin{array}{*{20}{c}} {{u^{\prime\prime}} + g\left( {t,u,{u^{\prime}}} \right) = 0\quad {\text{for}}\quad t \in \left( {0,1} \right),} \hfill \\ {u(0) = u(1) = 0} \hfill \\ \end{array} } \right. $

is studied. The singularity may appear at u?=?0, and the function g may change sign. An existence theorem for solutions to the above boundary-value problem is proposed, and it is proved via the method of upper and lower solutions.

     

Profile of Blow-up Solution to Hyperbolic System with Nonlocal Term

This paper is concerned with a nonlocal hyperbolic system as follows utt = △u ＋ （∫Ωvdx ）^p for x∈R^N,t〉0 ,utt = △u ＋ （∫Ωvdx ）^q for x∈R^N,t〉0 ,u（x,0）=u0（x）,ut（x,0）=u01（x） for x∈R^N,u（x,0）=u0（x）,ut（x,0）=u01（x） for x∈R^N, where 1≤ N ≤3, p ≥1, q ≥ 1 and pq 〉 1. Here the initial values are compactly supported and Ω belong to R^N is a bounded open region. The blow-up curve, blow-up rate and profile of the solution are discussed.     

A survey and some new results on the existence of solutions of IPBVPS for first order functional differential equations

This paper deals with the periodic boundary value problem for nonlinear impulsive functional differential equation

Generalized Knopp identities for homogeneous Hardy sums and Cochrane-Hardy sums

Let q, h, a, b be integers with q > 0. The classical and the homogeneous Dedekind sums are defined by $$s(h,q) = \sum\limits_{j = 1}^q {\left( {\left( {{j \over q}} \right)} \right)\left( {\left( {{{hj} \over q}} \right)} \right),{\rm{ }}s(a,b,q) = \sum\limits_{j = 1}^q {\left( {\left( {{{aj} \over q}} \right)} \right)\left( {\left( {{{bj} \over q}} \right)} \right),} } $$ respectively, where $((x)) = \left\{ \begin{gathered} x - [x] - \tfrac{1} {2},if x is not an integer; \hfill \\ 0,if x is an integer. \hfill \\ \end{gathered} \right. $ The Knopp identities for the classical and the homogeneous Dedekind sum were the following: $$\sum\limits_{d|n} {\sum\limits_{r = 1}^d {s\left( {{n \over d}a + rq,dq} \right) = \sigma (n)s(a,q),} } $$ $$\sum\limits_{d|n} {\sum\limits_{{r_1} = 1}^d {\sum\limits_{{r_2} = 1}^d s \left( {{n \over d}a + {r_1}q,{n \over d}b + {r_2}q,dq} \right) = n\sigma (n)s(a,b,q),} } $$ where σ(n) =Σ _d|n d. In this paper generalized homogeneous Hardy sums and Cochrane-Hardy sums are defined, and their arithmetic properties are studied. Generalized Knopp identities for homogeneous Hardy sums and Cochrane-Hardy sums are given.     

具有脉冲的二阶三点边值问题存在性定理

In this paper, two existence theorems are given concerning the following 3-point boundary value problem of second order differential systems with impulses     

Global Existence and Blow-up of Solutions for Parabolic Systems Involving Cross-Diffusions and Nonlinear Boundary Conditions

Abstract In this paper, we investigate the positive solutions of strongly coupled nonlinear parabolic systems with nonlinear boundary conditions： {ut-a（u, v）△u=g（u, v）, vt-b（u, v）△v=h（u, v）, δu/δη=d（u, v）, δu/δη=f（u, v）.Under appropriate hypotheses on the functions a, b, g, h, d and f, we obtain that the solutions may exist globally or blow up in finite time by utilizing upper and lower solution techniques.     

Multiple Boundary Concentrating Solutions to Dirichlet Problem of Hénon Equation

Abstract  Let Ω be the unit ball centered at the origin in . We study the following problem

A Pohozaev Identity for the Fractional Hnon System

In this paper, we study the Pohozaev identity associated with a Henon-Lane-Emden system involving the fractional Laplacian:■in a star-shaped and bounded domain Ω for s ∈(0,1). As an application of our identity, we deduce the nonexistence of positive solutions in the critical and supercritical cases.     

The Existence of Solutions of Elliptic Equations with Neumann Boundary Condition for Superlinear Problems

In this paper, we study and discuss the existence of multiple solutions of a class of non-linear elliptic equations with Neumann boundary condition, and obtain at least seven non-trivial solutions in which two are positive, two are negative and three are sign-changing. The study of problem (1.1):{-△u αu=f(u),x∈Ω, x∈Ω,δu/δr=0,x∈δΩ,is based on the variational methods and critical point theory. We form our conclusion by using the sub-sup solution method, Mountain Pass Theorem in order intervals, Leray-Schauder degree theory and the invariance of decreasing flow.     

Ground states for Schrödinger–Poisson type systems

In this paper we consider the following elliptic system in \mathbbR³{\mathbb{R}^3}

Exact Non-Self-Similar Solutions of the Equation $$u_t = \Delta \ln u$$

In this paper, we obtain new exact non-self-similar solutions of the nonlinear diffusion equation $$\begin{gathered} {\text{ }}u_t = \Delta \ln u, \hfill \\ u \triangleq u\left( {x,t} \right):\Omega \times \mathbb{R}^ + \to \mathbb{R},{\text{ }} x \in \mathbb{R}^n , \hfill \\ \end{gathered} $$ where $\Omega \subset \mathbb{R}^n $ is the domain and $\mathbb{R}^ + = \left\{ {t:0 \leqslant t < + \infty } \right\},{\text{ }}u\left( {x,t} \right) \geqslant 0$ is the temperature of the medium.     

Cauchy problem for a semilinear Éidel’man parabolic equation

We establish conditions for the existence and uniqueness of a generalized solution of the Cauchy problem for the equation

Oscillatory properties of solutions of three-dimensional differential systems of neutral type

The purpose of this paper is to obtain oscillation criteria for the differential system

A relaxation result for micromagnetics in SBV

An integral representation for the functional

Weighted Elliptic Systems of Lane–Emden Type in Unbounded Domains

We study the existence of weak solutions for a nonlinear elliptic system of Lane-Emden type $$\left\{\begin{array}{ll} -\Delta u \; = \; {\rm sgn}(v)|v|^{p-1} & {\rm in}\;\mathbb{R}^N, \\ -\Delta v \; = \; -\rho(x){\rm sgn}(u)|u|^{\frac{1}{p-1}} + f(x, u) & {\rm in}\;\mathbb{R}^N, \\ u, v \to 0 \quad {\rm as} \quad |x| \to +\infty, \end{array}\right.$$ by means of the Mountain Pass Theorem and some compact imbeddings in weighted Sobolev spaces.     

Approximating periodic functions in the uniform metric by Jackson type polynomials

Let C be the space of continuous 2π-periodic functions f with the norm . Let , where , be the Jackson polynomials of a function f, E _n (f) be the best approximation of f in the space C by trigonometric polynomials of order n, and let , be the function trigonometrically conjugate to the primitive of f. The paper establishes results of the following types: