Sharp well-posedness of the cauchy problem for the higher-order dispersive equation

This current paper is devoted to the Cauchy problem for higher order dispersive equation u_t+ ?_x~(2n+1)u = ?_x(u?_x~nu) + ?_x~(n-1)(u_x~2), n ≥ 2, n ∈ N~+.By using Besov-type spaces, we prove that the associated problem is locally well-posed in H~(-n/2+3/4,-1/(2n))(R). The new ingredient is that we establish some new dyadic bilinear estimates. When n is even, we also prove that the associated equation is ill-posed in H~(s,a)(R) with s -n/2+3/4 and all a∈R.     

Small data solutions of 2-D quasilinear wave equations under null conditions

2 Abstract For the 2-D quasilinear wave equation(?_t~2-?_x)u+2∑ij=0g~(ij)(?u)?_(ij)u = 0 satisfying i,j=0 null condition or both null conditions, a blowup or global existence result has been shown by Alinhac. In this paper, we consider a more general 2-D quasilinear wave equation(?_t~2-?_x)_u+2∑ij=0g~(ij)(?u)?_(ij)u = 0 satisfying null conditions with small initial data and the coefficients i,j=0 depending simultaneously on u and ?u. Through construction of an approximate solution,combined with weighted energy integral method, a quasi-global or global existence solution are established by continuous induction.     

利普希兹区域上薛定谔方程Neumann问题的加权估计

该文研究了利普希兹区域上加权空间H~p(?Ω,ω_αdσ)和L~p(?Ω,ω_αdσ)(1-εp≤2)上薛定谔方程-△u+Vu=0加权估计问题.记Ω是R~n(n≥3)上边界连通的有界利普希兹区域.令ω_α(Q)=|Q-Q_0|~α,这里Q_0是?Ω上的一个不动点.对于:定义在Ω上的薛定谔方程-△u+Vu=0,其中奇异非负位势V属于反H?lder类-B_n.该文研究边值落在加权空间H~p(?Ω,ω_αdσ)或L~p(?Ω,ω_αdσ)上的Neumann问题,这里dσ表示?Ω上的测度.对于特定范围的α,方程存在唯一解u,使得非切向的极大函数▽u在H~p(?Ω,ω_αdσ)或L~p(?Ω,ω_αdσ)上.此外,还建立了这些解的一致估计.     

Fokas方程的显示孤立尖波解和周期尖波解

利用一个独立变换和动力系统方法对Fokas方程:u_(tx)=(1+v(?)■_x~2)sin(u),x∈R,t0进行研究.在对该方程所对应的平面动力系统进行定性分析的基础上,得到了该方程所有可能的显式孤立尖波解和周期尖波解.     

Banach空间四阶周期边值问题正解的存在性

讨论了Banach空间E中的四阶周期边值问题:( u(4)(t)??u00(t)+′u(t)= f(t; u(t));06 t 61; u(i)(0)= u(i)(1); i =0;1;2;3正解的存在性,其中f :[0;1]￡ P ! P连续, P为E的正元锥,?;′2 R且满足0<′<(?2+2?2)2;?>?2?2;′?4+??2+1>0:通过非紧性测度的估计技巧与凝聚映射的不动点指数理论获得了该问题正解的存在性结果.     

Critical Trace Trudinger-Moser Inequalities on a Compact Riemann Surface with Smooth Boundary

In this paper, the author concerns two trace Trudinger-Moser inequalities and obtains the corresponding extremal functions on a compact Riemann surface (Σ, g) with smooth boundary ?Σ. Explicitly, let λ₁(?Σ) = inf u∈W^1,2 (Σ,g),R ?Σ udsg=0,u6≡0 R Σ(|?gu|2 + u2 )dvg R ?Σ u2 dsg and H = n u ∈ W^1,2 (Σ, g) : Z Σ(|?gu|2 + u2 )dvg ? α Z ?Σ u2dsg ≤ 1 and Z ?Σ u dsg = 0o ,where W^1,2 (Σ, g) denotes the usual Sobolev space and ?g stands for the gradient operator.By the method of blow-up analysis, we obtain sup u∈H Z ?Σ e πu2 dsg ( < +∞, 0 ≤ α < λ1(?Σ),= +∞, α ≥ λ1(?Σ).Moreover, the author proves the above supremum is attained by a function uα ∈ H∩C∞(Σ)for any 0 ≤ α < λ1(?Σ). Further, he extends the result to the case of higher order eigenvalues. The results generalize those of [Li, Y. and Liu, P., Moser-Trudinger inequality on the boundary of compact Riemannian surface, Math. Z., 250, 2005, 363–386], [Yang,Y., Moser-Trudinger trace inequalities on a compact Riemannian surface with boundary,Pacific J. Math., 227, 2006, 177–200] and [Yang, Y., Extremal functions for TrudingerMoser inequalities of Adimurthi-Druet type in dimension two, J. Diff. Eq., 258, 2015,3161–3193]     

具有较高级算子的两组分Camassa-Holm方程的柯西问题

讨论了具有高阶算子-(1-_x~2+_x~4)的两组分Camassa-Holm方程在Sobolev空间H~s,s>5/2中的柯西问题,利用构造逼近解方法证明了该方程解的局部适定性问题,同时应用能量估计及嵌入定理等方法得到了在给定初值条件下的爆破准则,并且进一步给出了具体的爆破速率.     

广义Kawahara方程的Cauchy问题

对初值在Besov空间中的广义Kawahara方程(?)_tu αu~k(?)_xu β(?)_x~3u γ(?)_x~5u=0进行了研究,其中k是大于4的正整数,证明了对任意的1≤q≤∞,其Cauchy问题在Besov空间B_(2,q)~(sk)(R)和B_(2,q)~s(R)中局部适定,这里s_k=(k-8)/2k,s＞max(0,s_k)；对小初值问题几乎整体适定．并证明了如果β=0或βγ＜0,对小初值问题整体适定．     

A tracing of the fractional temperature field

This paper is devoted to a study of L~q-tracing of the fractional temperature field u(t, x)—the weak solution of the fractional heat equation(?_t +(-?_x)~α)u(t, x) = g(t, x) in L~p(R_+~(1+n)) subject to the initial temperature u(0, x) = f(x) in L~p(R~n).     

有限图上高阶Yamabe型方程的非平凡解

该文研究了以下高阶Yamabe型方程Lm,pu?g|u|^p?2u=λf|u|α?2u在有限图上的非平凡正解的存在性,其中Lm,p是一个2m阶差分算子,它是一种p次(?Δ)^m算子更一般化,α≥p≥2,g>0和f>0是定义在G的所有顶点上的实函数,m≥1是一个整数.     

环F_2+uF_2上长为2~e的(1+u)-循环码

最近,环F2+uF2上的线性码引起了编码研究者极大的兴趣.本文证明了R[x]/〈xn+1+u〉是有限链环,其中R=F2+uF2=F2[u]/〈u2〉且n=2e.从而给出了F2+uF2上的所有长为2e的(1+u)-循环码,进而给出了所有(1+u)-循环码的对偶码.证明了F2+uF2上不存在长为2e的非平凡的自对偶的(1+u)-循环码.     

环F_2+uF_2+vF_2+uvF_2上长度为2~s的常循环码（英文）

本文研究了环R=F_2+uF_2+vF_2+uvF_2上长度为2~s的常循环码的分类和结构,这个环是一个局部环,但不是链环.首先,借助有限交换局部环中多项式的欧几里德算法,得到了长为2~s的循环码与(1+uv)-常循环码分类,且给出了每一类的结构.其次,利用(x-1)~(2~s)=u,得到了长为2~s的(1+u)-常循环码分类和每一类的结构.最后,利用类似于长为2~s的(1+u)-常循环码的讨论方法,给出了(1+v),(1+u+uv),(1+v+uv),(1+u+v),(1+u+v+uv)-常循环码分类和每一类的结构.     

关于一个积分不等式的一些注记

Given two positive constantsαandβ,we prove that the integral inequality∫_0~1 f~(α+β)(x)dx≥∫_0~1 f~α(x)x~βdx holds for all non-negative valued continuous functions f satisfying∫_x~1 f(t)dt≥∫_x~1 tdt for x∈[0,1]if and only ifα+β≥1.This solves an open problem proposed recently by Ngo,Thang,Dat,and Tuan.     

Liouville theorems for an integral system with Poisson kernel on the upper half space

We investigate the Liouville theorem for an integral system with Poisson kernel on the upper half space R_+~n,{u(x) =2/(nω_n)∫_(?R_+~n)(xnf(v(y)))/(|x- y|~n)dy, x ∈R_+~n,v(y) =2/(nω_n)∫_(R_+~n)(xng(u(x)))/(|x- y|~n)dx, y ∈?R_+~n,where n 3, ωn is the volume of the unit ball in Rn. This integral system arises from the Euler-Lagrange equation corresponding to an integral inequality on the upper half space established by Hang et al.(2008).With natural structure conditions on f and g, we classify the positive solutions of the above system based on the method of moving spheres in integral form and the inequality mentioned above.     

环F_2+uF_2上偶长的(1+u)-常循环码

给出了环F2+uF2上任意偶长的(1+u)-常循环码的结构,确定了给定偶长度F2+uF2上(1+u)-常循环码的数目.通过Gray映射,得到了F2+uF2上偶长的(1+u)-常循环码的二元象.     

MULTIPLICITY RESULTS FOR AN INHOMOGENEOUS NONLINEAR ELLIPTIC PROBLEM

The author proves that there exist three solutions u0,u1 and u2 in the following problem {-△u+u=Q(x)|u|P-2u+f(x),u∈H1(RN),2＜P＜2u,2u=2N/N-2,N≥3,where some conditions are imposed on Q and f. Here,0＜u0＜u1,u2 changes sign.     

奇数阶边值问题正解的存在性与多重性

利用锥拉伸锥压缩不动点定理,证明了在一定条件下,下列非线性奇数阶方程(-1)q+1u(2q+1)(t)=λa(t)f(u(t)),0 t 1,(-1)q+1u(2q+1)(t)=λa(t)f(u(t)),0 t 1,u(0)=u′(τ)=u″(1)=0u(2j+1)(0)=u(2j+1)(1)=0,j=1,2,…,q-1.单个和多个正解的存在性,其中λ>0,12<τ<1,q∈N.得到了λ的区间Λ,对一切λ∈Λ,该问题至少有一个正解,同样也得到了该问题至少有两个正解λ相应的区间.     

带有临界指标的周期Schrdinger方程的基态解

该文证明周期Schrdinger方程-△u+V(x)u=K(x)|u|~(2*-2)u+g(x,u),u∈H~1(R~N)基态解的存在性,其中N4,2~*=(2N)/(N-2)为临界Sobolev指标.该文补充了以上方程关于基态解存在性的以往结果.     

一道微分方程题目的多种解法

同济大学数学教研室编高等数学 (第四版 )下册 P40 7有一题目 :求方程的通解。学生普遍感到有些困难。下面给出几种解法。y′+x =x2 +y ( 1 )　　解　方法一　令 x2 +y-x=u,则 yx2 +y+x=u,y=u( x2 +y+x) ,两边对 x求导 ,得 dydx= ( x2 +y+x) dudx+u(2 x+dydx2 x2 +y+1 )。代入 ( 1 ) ,得 dudx+u2 ( u+x) =0 ,或udx +2 ( u +x) du =0 ( 2 )易见有积分因子 μ=u,引用之 ,解得 2 u3 +3 xu2 =c1。换回原变量 ,得 ( 1 )的通解为 ( x2 +y) 3 =x3 +32 xy+c.其中 c=c12 为任意常数。方法二　令 u=x2 +yx ,则 x2 +y =ux,两边对 x求导 ,得2 x+dydx2 x…     

带有临界增长的Kirchhoff型问题的基态解

本文主要考虑如下Kirchhoff问题{-(a+b∫R_3|?u|~2dx)?u+u=f(x,u)+Q(x)|u|~4u,u∈H~1(R~3),其中a,b是正的常数.我们证明了基态解,即上述问题的极小能量解的存在性.同时,如果假定Q≡1,且h(x)满足一定的条件,可以证明下述问题{-(a+b∫R_3|?u|~2dx)?u+u=|u|~4u+h(x)u,u∈H~1(R~3)的基态解的存在性.