On the existence of solutions for an elliptic system of equations with arbitrary order growth

Let BR be the ball centered at the origin with radius R in RN ( N ≥2). In this paper we study the existence of solution for the following elliptic systemu -△u+λu=p/(p + q)κ(| x |)) u(p-1)vq1,x ∈BR1,-△u+λu=p/(p + q)κ(| x |)) upv(q-1)1,x ∈BR1,u > 01,v > 01,x ∈ BR1,(u)/(v)=01,(v)/(v)=01,x ∈BRwhereλ > 0 , μ > 0 p ≥ 2, q ≥ 2,ν is the unit outward normal at the boundary BR . Under certainassumptions on κ ( | x | ), using variational methods, we prove the existence of a positive and radially increasing solution for this problem without growth conditions on the nonlinearity.     

变时间分数阶扩散方程的非一致时间网格有限差分方法

本文在非一致时间网格上,使用有限差分方法求解变时间分数阶扩散方程?α(x,t)u(x,t)/tα(x,t)-2u(x,t)/x2=f(x,t),0α(x,t)q≤1,证明了该方法在最大范数下的稳定性与收敛性,收敛阶为C(Δt2-q+h2).数值实例验证了理论分析的结果.     

一类非线性抛物方程组解的爆破时间上下界估计

本文研究了一类非线性抛物方程组uj/t=△uj+fj(u)解的爆破时间的估计问题.通过构造恰当的辅助函数和建立一系列微分不等式,获得了此类非线性抛物方程组解的爆破时间上下界的估计.从而将单个方程的结论推广到了方程组的情形.     

EXISTENCE AND UNIQUENESS OFWEAK SOLUTIONS OF UNIFORMLY DEGENERATE QUASILINEAR PARABOLIC EQUATIONS

In this paper we deal with the quasilinear parabolic equation u/t=/x_i[a_(ij)(x, t, u))u/x_j]+b_i(x, t, u)u/x_i+c(x, t, u) which is uniformly degenerate at u=O. Under some assumptions we prove existence anduniqueness of nonnegative weak solutions to the Cauchy problem and the first boundary valueproblem for this equation. Furthermore, the weak solutions are globally Holder continuous.     

On the permutability of Backlund transformations

Let(?)=B_ηu:2(q-(?))+(⊿((?)-2q))+(2q_x+(?)_x))η=0,2(r-(?)+(⊿(2(?)-r)+(r_x+2(?)_x))η=0,u=(q,r)~Tbe the Backlund transformation (BT) of the hierarchy of AKNS equations,where η is a parameterand Δ=integral from -∞ to x (qr-(?))dx′.It is shown in this paper the infinitesimal BT B_(η+ε)B_η~(-1) admits thefollowing expansionB_(η+ε)B_η~(-1)u=u+εsum from n=0 to ∞ β_n(JL~(n+1)u)η~n,β_n=1+(-1)~n2~(-n-1),where L is the recurrence operator of the hierarchy and ε is an infinitesimal parameter.Thisexpansion implies the equivalence between the permutabiliy of BTs and the involution in pairs ofconserved densities.     

奇异二维p-Laplacian方程多点边值问题正解的存在性

本文研究具有p-Laplacian算子的奇异多点边值问题{（Фp（u＇））＇＋q（t）f（t,u）=0,0〈t〈1;u（0）=∑n i=1 αiu（ηi）,u（i）∑n i=1 βiu（ηi）正解的存在性，其中f（t，u）可以在u=0奇异，q（t）可以在t=0或t=1奇异．     

本征值问题的统一推导方法及讨论

针对一般情形的本征方程X″(x)-2bX′(x)+λX(x)=0结合第一、二、三类齐次边界条件的统一形式,给出有关本征值问题的统一结果,从而可直接利用分离变量法求解2U/t2=a22U/x2+a1u/x+a2t/u+a3u型等含有ux项的泛定方程的定解问题.     

平面拟共形映照的偏差函数估计

研究平面拟共形映照的偏差函数μ(r),λ(K)和ηK(t).利用环形区域模函数的共形不变性,证明μ(r)满足一个新的不等式.作为应用,不必利用椭圆函数的性质,得到了估计Gr(o|¨)tzsch,Teichm(u|¨)ller和Mori这3种典型极值环形区域模函数的更精确的不等式,并得到了λ(K)和ηK(t)的更精确的上下界估计不等式.改进了由Anderson,Vamanamurthy,Qiu和Vuorinen所得的相应结果.     

一类强退化抛物方程弱解的惟一性

研究了如下形式的强退化抛物方程(C)(u)/(t)=(2A(u,x,t))/(x2)+(B(u,x,t))/(x),基于Holm gren方法,证明了弱解的惟一性.     

关于广义Hardy-Hilbert积分不等式及其应用

通过引入参数λ(1-q/p＜λ≤2,p≥q＞1)及两个非负且在(0, ∞)递增的可微函数u(x)和v(x)建立了一种广义带权的Hardy-Hilbert积分不等式.特别,当p=2时,得到经典Hilbert积分不等式的各种推广.作为应用,当u(x)和v(x)是幂函数、指数函数和对数函数时,建立了若干重要不等式.     

有界域上具有离散全纯核的Koppelman公式与方程

D是C^n空间中具有逐块C(1)边界的有界域，该文建立了D上一个具有离散局部全纯核的(0，q)形式的Koppelman积分公式及其相应的方程解的积分表示和它的内闭一致估计式。     

七阶非线性色散方程初值问题解的局部和整体存在性

该文研究七阶非线性弱色散方程:∂u/∂t + au(∂u/∂x) +β(∂^3 u/∂x^3) +γ(∂^5 u/∂x^5) + μ(∂^7 u/∂x^7)=0, (x,t)∈R^2的初值问题,通过运用震荡积分衰减估计的最近结果, 首先对相应线性方程的基本解建立了几类Strichartz型估计. 其次, 应用这些估计证明了七阶非线性弱色散方程初值问题解的局部与整体存在性和唯一性. 结果表明, 当初值u_0(x)∈H^s(R), s≥2/13 时, 存在局部解; 当s≥1时, 存在整体解.     

一类带反应项的 Othmer-Stevens型趋化模型解的存在性

该文讨论了一类带反应项的Othmer-Stevens 型趋化模型的初边值问题 {∂u/∂t=D∨(u∨lnu/Φ(x, t, w))+ f(x, t, u), ∂w/∂t=g(x, t, u, w), u∨lnu/Φ(x, t, w) ?n^→=0. 证明了: 如果边界∂Ω ∈C^2+β, 函数Φ(x, t , w), f(x, t, u) 和 g(x, t, u, w)充分光滑,则该系统存在唯一解.     

Projective Dirichlet boundary condition with applications to a geometric problem

Given a domain Ω ? R~n, let λ 0 be an eigenvalue of the elliptic operator L :=Σ!(i,j)~n =1?/?xi(a~(ij0 ?/?xj) on Ω for Dirichlet condition. For a function f ∈ L~2(Ω), it is known that the linear resonance equation Lu + λu = f in Ω with Dirichlet boundary condition is not always solvable.We give a new boundary condition P_λ(u|? Ω) = g, called to be pro jective Dirichlet condition, such that the linear resonance equation always admits a unique solution u being orthogonal to all of the eigenfunctions corresponding to λ which satisfies ||u||2,2 ≤ C(||f ||_2 +|| g||_(2,2)) under suitable regularity assumptions on ?Ω and L, where C is a constant depends only on n, Ω, and L. More a priori estimates,such as W~(2,p)-estimates and the C~(2,α)-estimates etc., are given also. This boundary condition can be viewed as a generalization of the Dirichlet condition to resonance equations and shows its advantage when applying to nonlinear resonance equations. In particular, this enables us to find the new indicatrices with vanishing mean(Cartan) torsion in Minkowski geometry. It is known that the geometry of indicatries is the foundation of Finsler geometry.     

一个非线性色散波方程的惟一连续性

该文证明:和如下耦合色散系统相联系的初值问题的充分光滑的解$(u,v)=(u(x,t),v(x,t))$, 如果在两个时刻有半线支集那么它们全为零. {∂ _tu+∂³_x u+∂ _x(u^p v^p+1)=0, ∂ _tv+∂³_x v+∂_x(u^p+1v^p)=0,x∈R,t≥ 0     

一类奇异非线性三点边值问题的正解

应用锥上的不动点定理，建立了奇异非线性三点边值问题(u″(t)+a(t)f(u)=0,0＜t＜1,αu(0)-βu′(0)=0,u(1)-ku(η)=0)正解的一个存在性定理．这里η∈(0,1)是一个常数，a∈C( (0,1),［0,+∞)),f∈C(［0,+∞),［0,+∞))     

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.     

一类次P-Laplace方程非负解的不存在性

该文得到了在Ω上以下问题 {L_p,_ku+f(u)=0, , u|∂Ω=0 非负解的不存在性结果. 其中Ω为Heisenberg型群G中的区域(有界或无界), L_{p, k}u=divX (| _X u|^p-2 _X u)为对应于Greiner型向量场 X 的一类次P-Laplace算子.     

A survey of pseudo (v,k, λ)-designs

This work is an attempt to give a complete survey of all known results about pseudo (v, k, )-designs. In doing this, the author hopes to bring more attention to his conjecture given in Section 6; an affirmative answer to this conjecture would settle completely the existence and construction problem for a pseudo (v, k, )-design in terms of the existence of an appropriate (v, k, )-design.     

On the Characterisation of AG(n,q) by its Parameters asa Nearly Triply Regular Design

We show that a non-symmetric nearly triply regular designD with and in which every line has at least q points is AG(n,q) for prime power q > 2 and positiveinteger n 3.