本刊英文版Vol.32(2016),No.5论文摘要（英文）

正Long Time Dynamics of the 3D Radial NLS with the Combined Terms Gui Xiang XU Jian Wei Yang Abstract In this paper,we study the scattering and blow-up dichotomy result of the radial solution to nonlinear Schr(o|¨)dinger equation(NLS)with the combined terms iu_t+△u=-|u|~4u+|u|~(p-1)u,1+4/3p5in energy space H~1(R~3).The threshold energy is the energy of the ground state W of the focusing,energy critical NLS,which means that the subcritical perturbation does not affect the determination of threshold,but affects the scattering and blow-up dichotomy result with     

Infinitely Many Solutions for Schr?dinger–Choquard–Kirchhoff Equations Involving the Fractional p-Laplacian

In this article,we show the existence of infinitely many solutions for the fractional pLaplacian equations of Schr?dinger-Kirchhoff type equation ■ ,where(-△)_p~s is the fractional p-Laplacian operator,[u]_(s,p) is the Gagliardo p-seminorm,0 s 1 q p N/s,α∈(0,N),M and V are continuous and positive functions,and k(x) is a non-negative function in an appropriate Lebesgue space.Combining the concentration-compactness principle in fractional Sobolev space and Kajikiya's new version of the symmetric mountain pass lemma,we obtain the existence of infinitely many solutions which tend to zero for suitable positive parameters λ and β.     

广义Davey-Stewartson系统驻波的强不稳定性

本文研究方程驻波的强不稳定性iu_t+△u+a|u|~(p-1)u+E_1(|u|~2)u=0,t≥0,x∈R~n,其中a0,1p(n+2)/(n+2)~+,n∈{2,3}.当1+4/n≤pn+2/(n-2)~+)时,文[Sharp threshold of global existence and instability of standing wave for a Davey-Stewartson system,Commun.Math.Phys.,2008,283:93-125]在驻波的频率满足一定假设条件下,证明了此方程驻波的强不稳定性.本文去掉这个假设,得到相同的结论.     

临界或超临界增长分数阶Schrodinger-Poisson方程正解的存在性

本文研究如下分数阶Schrodinger-Poisson方程{(-△)^su+Vx(u)+K(x)φu=f(u)+λ|u|^q-2ux∈R³,(-△)^tφ=K(x)u²,x∈R³其中S∈(3/4,1),t∈(0,1),f是在原点超线性无穷远次临界的连续非线性项,指数q≥2_s^*=6/3-2x.当λ>0充分小时,我们利用变分方法证明上述问题正解的存在性.本文的主要贡献是处理了超临界情形.     

次临界分数阶Laplace方程具有两个bubbles的变号解存在性

考虑次临界分数阶Laplace问题{(-△)~su=︳u︳~(p-1-ε)u,x∈Ω,u=0,x∈?Ω}具有两个bubbles的变号解的存在性,其中Ω是R~N中的有界光滑区域,N2s,0s1,p=(N+2s)/(N-2s),ε0充分小.这个工作可以看作Bartsch,Micheletti,Pistoia在文[On the existence and the profile of nodal solutions of elliptic equations involving critical growth,Calc.Var.Partial Differential Equations,2006,3:265-282]结果的一种非局部形式的推广.     

带非奇扰动项的(2,p)-Laplace方程无穷多解的存在性

本文研究带非奇扰动项的(2,p)-Laplace方程{u=0,-△u-△pu=a(x)|u|^q-2u+f(x,u)x∈ЭΩ,x∈Ω,其中ΩСR^N是有界光滑区域,1相似文献   

一类Kirchhoff方程Neumann边值问题解的存在性

考虑如下一类Kirchhoff方程Neumann边值问题:{-(a+b∫Ω(|↓△u|²+|u|²dx)(△u-u)+=c(x)|u|^q-2u+f(x,u)■u/■v=0,其中Ω■R^N是光滑有界域,c(x)可能是变号函数,a≥0,b>0且a+b>0,1相似文献   

一类不满足(AR)条件的p-Laplace方程的无穷多解

利用变分方法,得到以下p-Laplace方程-Δ_pu+V(x)|u|^p-2u=f(x,u),x∈R^N,(1)有无穷多高能量.其中1N上无界函数,非线性项f(x,u)不满足(AR)条件.     

Multiple Positive Solutions for a Nonlinear Elliptic Equation Involving Hardy–Sobolev–Maz'ya Term

In this paper, we study the existence and nonexistence of multiple positive solutions for the following problem involving Hardy–Sobolev–Maz'ya term:-Δu- λu/|y|2=|u|pt-1u/|y|t+ μf(x), x ∈Ω,where Ω is a bounded domain in RN(N ≥ 2), 0 ∈Ω, x =(y, z) ∈ Rk× RN-kand pt =N +2-2t N-2(0 ≤ t ≤2). For f(x) ∈ C1(Ω)\{0}, we show that there exists a constant μ* 0 such that the problem possessesat least two positive solutions if μ∈(0, μ*) and at least one positive solution if μ = μ*. Furthermore,there are no positive solutions if μ∈(μ*, +∞).     

Bounds for the Blow-Up Time on the Pseudo-Parabolic Equation with Nonlocal Term

We investigate the initial boundary value problem of the pseudo-parabolic equation $u_{t} - \triangle u_{t} - \triangle u = \phi_{u}u + |u|^{p - 1}u,$ where $\phi_{u}$ is the Newtonian potential, which was studied by Zhu et al. (Appl. Math. Comput., 329 (2018) 38-51), and the global existence and the finite time blow-up of the solutions were studied by the potential well method under the subcritical and critical initial energy levels. We in this note determine the upper and lower bounds for the blow-up time. While estimating the upper bound of blow-up time, we also find a sufficient condition of the solution blowing-up in finite time at arbitrary initial energy level. Moreover, we also refine the upper bounds for the blow-up time under the negative initial energy.     

Existence of semiclassical states for a quasilinear Schrödinger equation on ℝ<Superscript><Emphasis Type="Italic">N</Emphasis></Superscript> with exponential critical growth

We study a quasilinear Schrödinger equation -ε^NΔ_Nu+V(x)|u|^N-2u=Q(x)f(u) in R^N, 01,N(R^N),u(x)0, where V, Q are two continuous real functions on R^N and ε>0 is a real parameter. Assume that the nonlinearity f is of exponential critical growth in the sense of Trudinger-Moser inequality, we are able to establish the existence and concentration of the semiclassical solutions by variational methods.     

本刊英文版Vol.30(2014),No.4论文摘要

<正>The Defocusing Energy-supercritical Hartree Equation Ji Qiang ZHENGAbstract In this paper,we study the global well-posedness and scattering problem for the energy-supercritical Hartree equation iu_t+△u-(|x|~(-γ)*|u|~2)u=0 withγ4 in dimension dγ.We prove that if the solution u is apriorily bounded in the critical Sobolev space,that is,u∈L_t~∞(I;H_x~sc(R~d))with s_c:=γ/2-11,then u is global and scatters.The impetus to     

KAM Tori for the Derivative Quintic Nonlinear Schrodinger Equation

This paper is concerned with one-dimensional derivative quintic nonlinear Schrodinger equation,iut—uxx+i(|u|4u)x=0,x eT.The existence of a large amount of quasi-periodic solutions with two frequencies for this equation is established.The proof is based on partial Birkhoff normal form technique and an unbounded KAM theorem.We mention that in the present paper the mean value of u does not need to be zero,but small enough,which is different from the assumption(1.7)in Geng-Wu[J.Math.Phys.、53,102702(2012)].     

Kirchhoff方程单峰解的局部唯一性

该文主要证明了以下非线Kirchhoff问题的单峰解的局部唯一性-(∈^2a+∈b∫R^3|▽u|^2dx)△u+u=K(x)|u|^p-1u,u> 0,x∈R^3,其中∈>0任意小,a,b> 0,1相似文献   

Well-posedness,Decay Estimates and Blow-up Theorem for the Forced NLS

In this article we prove that the following NLS iu_t = u_{zz}-g|u|^{P-1}u,g > O, x, t > 0 with either Dirichlet or Robin boundary condition at x = 0 is well-posed. L^{p + 1} decay estimates, blow-up theorem and numerical results are also given.     

具结构阻尼的拟线性膜方程的吸引子及其上半连续性

本文研究具结构阻尼的拟线性膜方程utt+Δ²u+(-Δ)^αut+Δφ(Δu)+f(u)=g的适定性以及解的长时间动力学行为,其中α∈(1,2),旨在研究耗散指标α对方程解的适定性和长时间动力学行为的影响.本文证明非线性项φ(s)存在一个依赖于耗散指标α的临界指数pα=(N+4(α-1))/(N-4(α-1))⁺(N=3,4),当1≤pα时,对f(u)没有任何多项式增长限制:(ⅰ)方程的初边值问题是适定的,其解当t> 0时具有整体正则性;(ⅱ)对任意α∈(1,2),对应的解算子半群S^α(t)在自然能量空间中存在整体吸引子和指数吸引子;(ⅲ)整体吸引子族{Aα}在任意点α0∈(1,2)处上半连续,即对Aα0。的任意邻域U,当|α-α0|<<1时有Aα■U.     

Scattering in the energy space for the NLS with variable coefficients

We consider the NLS with variable coefficients in dimension \(n\ge 3\)

$$\begin{aligned} i \partial _t u - Lu +f(u)=0, \qquad Lv=\nabla ^{b}\cdot (a(x)\nabla ^{b}v)-c(x)v, \qquad \nabla ^{b}=\nabla +ib(x), \end{aligned}$$

on \(\mathbb {R}^{n}\) or more generally on an exterior domain with Dirichlet boundary conditions, for a gauge invariant, defocusing nonlinearity of power type \(f(u)\simeq |u|^{\gamma -1}u\). We assume that L is a small, long range perturbation of \(\Delta \), plus a potential with a large positive part. The first main result of the paper is a bilinear smoothing (interaction Morawetz) estimate for the solution. As an application, under the conditional assumption that Strichartz estimates are valid for the linear flow \(e^{itL}\), we prove global well posedness in the energy space for subcritical powers \(\gamma <1+\frac{4}{n-2}\), and scattering provided \(\gamma >1+\frac{4}{n}\). When the domain is \(\mathbb {R}^{n}\), by extending the Strichartz estimates due to Tataru (Am J Math 130(3):571–634, 2008), we prove that the conditional assumption is satisfied and deduce well posedness and scattering in the energy space.

     

Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation

We consider the critical nonlinear Schrödinger equation $iu_{t} = -\Delta u-|u|^{4/N}$ with initial condition u(0, x) = u₀.For u₀$\in$H¹, local existence in time of solutions on an interval [0, T) is known, and there exist finite time blow-up solutions, that is u₀ such that $\textrm{lim} _{t\uparrow T <+\infty}|\nabla u(t)|_{L^{2}}=+\infty$. This is the smallest power in the nonlinearity for which blow-up occurs, and is critical in this sense.The question we address is to control the blow-up rate from above for small (in a certain sense) blow-up solutions with negative energy. In a previous paper [MeR], we established some blow-up properties of (NLS) in the energy space which implied a control $|\nabla u(t)|_{L^{2}} \leq C \frac{|\ln(T-t)|^{N/4}}{\sqrt{T-t}}$ and removed the rate of the known explicit blow-up solutions which is $\frac{C}{T-t}$.In this paper, we prove the sharp upper bound expected from numerics as$|\nabla u(t)|_{L^{2}} \leq C \left(\frac{\ln|\ln(T-t)|}{T-t} \right)^{1/2}$by exhibiting the exact geometrical structure of dispersion for the problem.     

A BLOW-UP RESULT FOR A CLASS DOUBLY NONLINEAR PARABOLIC EQUATIONS WITH VARIABLE-EXPONENT NONLINEARITIES

This paper deals with the following doubly nonlinear parabolic equations(u + |u|~(r(x)-2)u)t-div(|?u|~(m(x)-2)?u) = |u|~(p(x)-2)u, where the exponents of nonlinearity r(x), m(x) and p(x) are given functions. Under some appropriate assumptions on the exponents of nonlinearity, and with certain initial data, a blow-up result is established with positive initial energy.     

一类积分方程组正解的对称性和单调性

本文讨论积分方程组（？）解的性质,其中G_α是α阶贝塞尔位势核,0≤β〈α（n-α＋β）/n,1/（q＋1）＋1/（r＋1）〉（n-α＋β）/n,1/（r＋1）＋1/（p＋1）〉（n-α＋β）/n.我们用积分形式的移动平面法证明上述积分方程组的正解是径向对称且单调的.