Existence of Generalized Heteroclinic Solutions of the Coupled Schr¨odinger System under a Small Perturbation

The following coupled Schrodinger system with a small perturbation
is considered, where β and ε are small parameters. The whole system has a periodic solution with the aid of a Fourier series expansion technique, and its dominant system has a heteroclinic solution. Then adjusting some appropriate constants and applying the fixed point theorem and the perturbation method yield that this heteroclinic solution deforms to a heteroclinic solution exponentially approaching the obtained periodic solution （called the generalized heteroclinic solution thereafter）.     

Homogenization of a parabolic equation in perforated domain with Neumann boundary condition

In this article, we study the homogenization of the family of parabolic equations over periodically perforated domains

Existence and concentration result for Kirchhoff equations with critical exponent and Hartree nonlinearity

This paper is concerned with the following Kirchhoff-type equations $$ \left\{ \begin{array}{ll} \displaystyle -\big(\varepsilon^{2}a+\varepsilon b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\mathrm{d}x\big)\Delta u + V(x)u+\mu\phi |u|^{p-2}u=f(x,u), &\quad \mbox{ in }\mathbb{R}^{3},\(-\Delta)^{\frac{\alpha}{2}} \phi=\mu|u|^{p},~u>0, &\quad \mbox{ in }\mathbb{R}^{3},\\end{array} \right. $$ where $f(x,u)=\lambda K(x)|u|^{q-2}u+Q(x)|u|^{4}u$, $a>0,~b,~\mu\geq0$ are constants, $\alpha\in(0,3)$, $p\in[2,3),~q\in[2p,6)$ and $\varepsilon,~\lambda>0$ are parameters. Under some mild conditions on $V(x),~K(x)$ and $Q(x)$, we prove that the above system possesses a ground state solution $u_{\varepsilon}$ with exponential decay at infinity for $\lambda>0$ and $\varepsilon$ small enough. Furthermore, $u_{\varepsilon}$ concentrates around a global minimum point of $V(x)$ as $\varepsilon\rightarrow0$. The methods used here are based on minimax theorems and the concentration-compactness principle of Lions. Our results generalize and improve those in Liu and Guo (Z Angew Math Phys 66: 747-769, 2015), Zhao and Zhao (Nonlinear Anal 70: 2150-2164, 2009) and some other related literature.     

Orbital stability of periodic traveling wave solutions to the generalized Long-Short wave equations

This paper investigates the orbital stability of periodic traveling wave solutions to the generalized Long-Short wave equations $\left\{\begin{array}{l}i\varepsilon_{t}+\varepsilon_{xx}=n\varepsilon+\alpha|\varepsilon|^{2}\varepsilon,\\n_{t}=(|\varepsilon|^{2})_{x}, x\in R.\end{array} \right.$ Firstly, we show that there exist a smooth curve of positive traveling wave solutions of dnoidal type with a fixed fundamental period $L$ for the generalized Long-Short wave equations. Then, combining the classical method proposed by Benjamin, Bona et al., and detailed spectral analysis given by using Lame equation and Floquet theory, we show that the dnoidal type periodic wave solution is orbitally stable by perturbations with period $L$. As the modulus of the Jacobian elliptic function $k\rightarrow 1$, we obtain the orbital stability results of solitary wave solution with zero asymptotic value for the generalized Long-Short equations. In particular, as $\alpha=0$, we can also obtain the orbital stability results of periodic wave solutions and solitary wave solutions for the long-short wave resonance equations. The results in the present paper improve and extend the previous stability results of long-shore wave equations and its extension equations.     

NA样本部分线性模型估计的强相合性

考虑回归模型:Y~((j))(x_(in),t_(in))=t_(in)β+g(x_(in))+σ_(in)e~((j))(x_(in)),1≤j≤m,1≤i≤n,其中σ_(in)~2=f(u_(in)),(x_(in),t_(in),u_(in))为固定非随机设计点列,β是未知待估参数,g(·)和f(·)是未知函数,误差{e~((j))(x_(in))}是均值为零的NA变量.给出基于g(·)和f(·)一类非参数估计的β的最小二乘估计和加权最小二乘估计,并在适当条件下得到了它们的强相合性.     

A NECESSARY AND SUFFICIENT CONDITION OF EXISTENCE OF GLOBAL SOLUTIONS FOR SOME NONLINEAR HYPERBOLIC EQUATIONS

ＡＮＥＣＥＳＳＡＲＹＡＮＤＳＵＦＦＩＣＩＥＮＴＣＯＮＤＩＴＩＯＮＯＦＥＸＩＳＴＥＮＣＥＯＦＧＬＯＢＡＬＳＯＬＵＴＩＯＮＳＦＯＲＳＯＭＥＮＯＮＬＩＮＥＡＲＨＹＰＥＲＢＯＬＩＣＥＱＵＡＴＩＯＮＳ￥ＺＨＡＮＧＱＵＡＮＤＥ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅ...     

有向跳图的生成欧拉子有向图

一个有向多重图D的跳图$J(D)$是一个顶点集为$D$的弧集,其中$(a,b)$是$J(D)$的一条弧当且仅当存在有向多重图$D$中的顶点$u_1$, $v_1$, $u_2$, $v_2$,使得$a=(u_1,v_1)$, $b=(u_2,v_2)$ 并且$v_1\neq u_2$.本文刻画了有向多重图类$\mathcal{H}_1$和$\mathcal{H}_2$,并证明了一个有向多重图$D$的跳图$J(D)$是强连通的当且仅当$D\not\in \mathcal{H}_1$.特别地, $J(D)$是弱连通的当且仅当$D\not\in \mathcal{H}_2$.进一步, 得到以下结果: (i) 存在有向多重图类$\mathcal{D}$使得有向多重图$D$的强连通跳图$J(D)$是强迹连通的当且仅当$D\not\in\mathcal{D}$. (ii) 每一个有向多重图$D$的强连通跳图$J(D)$是弱迹连通的,因此是超欧拉的. (iii) 每一个有向多重图D的弱连通跳图$J(D)$含有生成迹.     

On the limit cycles for a class of generalized Kukles differential systems

In this paper, we consider the limit cycles of a class of polynomial differential systems of the form $\dot{x}=-y, \hspace{0.2cm} \dot{y}=x-f(x)-g(x)y-h(x)y^{2}-l(x)y^{3},$ where $f(x)=\epsilon f_{1}(x)+\epsilon^{2}f_{2}(x),$ $g(x)=\epsilon g_{1}(x)+\epsilon^{2}g_{2}(x),$ $h(x)=\epsilon h_{1}(x)+\epsilon^{2}h_{2}(x)$ and $l(x)=\epsilon l_{1}(x)+\epsilon^{2}l_{2}(x)$ where $f_{k}(x),$ $g_{k}(x),$ $h_{k}(x)$ and $l_{k}(x)$ have degree $n_{1},$ $n_{2},$ $n_{3}$ and $n_{4},$ respectively for each $k=1,2,$ and $\varepsilon$ is a small parameter. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of the linear center $\dot{x}=-y,$ $\dot{y}=x$ using the averaging theory of first and second order.     

重调和方程非平凡解的存在性

该文主要研究$R^N(N>4)$上重调和方程\begin{eqnarray*}\left\{\begin{array}{ll} \Delta^2 u+\lambda u=\overline{f}(x,u);\\ \lim\limits_{|x|\rightarrow\infty}u(x)=0;\\u\in{H^2}(R^N),\hspace{0.1cm}x\in{R^N } \end{array}\right.\end{eqnarray*}的非平凡解的存在性.为了便于研究,将方程转化为$R^N(N>4)$ 上带有扰动项的重调和方程\begin{eqnarray*}\left\{\begin{array}{ll} \Delta^2 u+\lambda u=f(u)+\varepsilon g(x,u);\\ \lim\limits_{|x|\rightarrow\infty}u(x)=0;\\u\in{H^2}(R^N),\hspace{0.1cm}x\in{R^N } .\end{array}\right.\end{eqnarray*}并运用扰动方法进行研究(其中$f(u)=\lim\limits_{|x|\longrightarrow \infty}\overline{f}(x,u),\varepsilon g(x,u)=\overline{f}(x,u)-f(u),\varepsilon$为任意小常数),证明了在适当条件下上述问题非平凡解的存在性.     

Nonexistence of global solutions of a nonlinear hyperbolic system

Consider the initial value problem

with and . We show that there exists a bound such that if all nontrivial solutions with compact support blow up in finite time.

     

On the Asymptotic Behavior of Solutions of the Heat Equation with a Random,Long-Range Correlated Potential

In this note we investigate the asymptotic behavior of the solutions of the heat equation with random, fast oscillating potential

Global attractor for the weakly damped driven Schrödinger equation in $ H^2 (\mathbb{R}) $

We discuss the asymptotic behaviour of the Schrödinger equation ¶¶$ iu_{t} + u_{xx} +i\alpha u -k\sigma(|u|^{2})u\, = f, \;\; x \in \mathbb{R}, \;\; t \geq 0,\;\;\;\alpha,\;k>0 $ iu_{t} + u_{xx} +i\alpha u -k\sigma(|u|^{2})u\, = f, \;\; x \in \mathbb{R}, \;\; t \geq 0,\;\;\;\alpha,\;k>0 ¶¶ with the initial condition u(x,0) = u₀ (x) u(x,0) = u_0 (x) . We prove existence of a global attractor in\ H² (\mathbbR) H^2 (\mathbb{R}) , by using a decomposition of the semigroup in weighted Sobolev spaces to overcome the noncompactness of the classical Sobolev embeddings.     

Asymptotic Expansion of Solutions of Parabolic Equations with a Small Parameter

The heat equation with a small parameter, $\left( {1 + \varepsilon ^{ - m} \chi \left( {\frac{x}{\varepsilon }} \right)} \right)ut = u_{xx} $ , is considered, where ε ∈ (0, 1), m < 1 and χ is a finite function. A complete asymptotic expansion of the solution in powers ε is constructed.     

Existence of entire positive solutions for semilinear elliptic systems with gradient term

Under the Keller?COsserman condition on ${\Sigma_{j=1}^{2}f_{j}}$ , we show the existence of entire positive solutions for the semilinear elliptic system ${\Delta u_{1}+|\nabla u_{1}|=p_{1}(x)f_{1}(u_{1},u_{2}), \Delta u_{2}+|\nabla u_{2}|=p_{2}(x)f_{2}(u_{1},u_{2}),x \in \mathbb{R}^{N}}$ , where ${p_{j}(j=1, 2):\mathbb{R}^{N} \rightarrow [0,\infty)}$ are continuous functions.     

Periodic Solutions of Nonlinear Wave Equations with Dissipative Boundary Conditions

Applying Nash-Moser's implicit function theorem, the author proves the existence of periodic solution to nonlinear wave equation u_{tt} - u_{xx} + εg(t, x, u, u_t, u_x, u_{tt}, u_{tx}, u_{xx}) = 0 with a dissipative boundary condition, provided ε is sufficiently small.     

Uniqueness and existence of solutions for a singular system with nonlocal operator via perturbation method

In this work, we investigate the existence and the uniqueness of solutions for the nonlocal elliptic system involving a singular nonlinearity as follows: $$ \left\{\begin{array}{ll} (-\Delta_p)^su = a(x)|u|^{q-2}u +\frac{1-\alpha}{2-\alpha-\beta} c(x)|u|^{-\alpha}|v|^{1-\beta}, \quad \text{in }\Omega,\ (-\Delta_p)^s v= b(x)|v|^{q-2}v +\frac{1-\beta}{2-\alpha-\beta} c(x)|u|^{1-\alpha}|v|^{-\beta}, \quad \text{in }\Omega,\ u=v = 0 ,\;\;\mbox{ in }\,\mathbb{R}^N\setminus\Omega, \end{array} \right. $$ where $\Omega $ is a bounded domain in $\mathbb{R}^{n}$ with smooth boundary, $0<\alpha <1,$ $0<\beta <1,$ $2-\alpha -\beta 相似文献   

Asymptotics of solutions to the periodic problem for the nonlinear damped wave equation

We study large time asymptotic behavior of solutions to the periodic problem for the nonlinear damped wave equation

Chemotaxis-fluild 系统的全局弱解

We investigate the existence of the global weak solution to the coupled Chemotaxisfluid system ■in a bounded smooth domain ??R~2. Here, r≥0 and μ 0 are given constants,?Φ∈L~∞(?) and g∈L~2((0, T); L_σ~2(?)) are prescribed functions. We obtain the local existence of the weak solution of the system by using the Schauder fixed point theorem. Furthermore, we study the regularity estimate of this system. Utilizing the regularity estimates, we obtain that the coupled Chemotaxis-fluid system with the initial-boundary value problem possesses a global weak solution.     

A Nonhomogeneous Boundary Value Problem for the Boussinesq Equation on a Bounded Domain

In this paper, we study the well-posedness of an initial-boundary-value problem (IBVP) for the Boussinesq equation on a bounded domain,\begin{cases} &u_{tt}-u_{xx}+(u^2)_{xx}+u_{xxxx}=0,\quad x\in (0,1), \;\;t>0,\\ &u(x,0)=\varphi(x),\;\;\; u_t(x,0)=ψ(x),\\ &u(0,t)=h_1(t),\;\;\;u(1,t)=h_2(t),\;\;\;u_{xx}(0,t)=h_3(t),\;\;\;u_{xx}(1,t)=h_4(t).\\ \end{cases} It is shown that the IBVP is locally well-posed in the space $H^s (0,1)$ for any $s\geq 0$ with the initial data $\varphi,$ $\psi$ lie in $H^s(0,1)$ and $ H^{s-2}(0,1)$, respectively, and the naturally compatible boundary data $h_1,$ $h_2$ in the space $H_{loc}^{(s+1)/2}(\mathbb{R}^+)$, and $h_3 $, $h_4$ in the the space of $H_{loc}^{(s-1)/2}(\mathbb{R}^+)$ with optimal regularity.     

A Nekhoroshev Type Theorem for the Nonlinear Wave Equation in Gevrey Space

In this paper, the authors prove a Nekhoroshev type theorem for the nonlinear wave equation u_(tt)=u_(xx)-mu-f(u),x∈[0, π] in Gevrey space.