甩瓦累-布然平均逼近连续函数

<正> §1.总说§1.1 设 f(x)∈C_(2π),f(x)～a_0/2+sum form n=1 to ∞ a_ncosnx+b_nsin nx≡sum form n=0 to ∞ A_n(x)记 S_n(f,x)=sum form v=0 to n A_v(x).称σ_(n,p)(f,x)=1/p+1 sum form v=n-p to n S_v(f,x)为 f(x)的瓦累-布然平均.记△_u~kf(x)=sum form v=0 to k (-1)~v(?)f[x+(k-2v)u].称函数ω_k(f,t)=(?)|△~u_kf(x)|为 f(x)的 k 阶连续模.简记ω(f,t)=ω_1(f,t).假如 f(x)的共轭函数     

高阶半线性抛物型方程组的生命跨度

本文研究高阶半线性抛物型方程组{ut+(-△)mu=|u|p, (t,x)∈R1+×RN, ut+(-△)mν=|u|q, (t,x)∈R1+×RN,u(0,x)=u0(x),v(0,x)=uo(x),x∈RN,其中m,p,q＞1.利用试验函数方法,首先推导一些积分不等式,然后对方程组爆破解的生命跨度[0,T)给出估计.     

LOCAL WELL-POSEDNESS OF INTERACTION EQUATIONS FOR SHORT AND LONG DISPERSIVE WAVES

The well-posedness of the Cauchy problem for the system{iδtu+δx^2u=uv+|u|^2u,t,x∈IR，δtv+δxHδxv=δx|u|^2,u(0,x)=u0(x),v(0,x)=v0(x),is considered. It is proved that there exists a unique local solution (u(x,t), v(x,t))∈C（[0,T);H^s)&#215;C([0,T);Hs^-1/2) for any initial data (u0,v0)∈H^s(IR)&#215;H^s-1/2(IR)(s≥1/4) and the solution depends continuously on the initial data.     

Rn上耦合的非线性Schrodinger程的正基态解

利用Nehari流形的方法和集中紧性引理,我们证明了Schrdinger系统{-△u+(1+a(x))u=Fu(u,v)+λv,-△v+(1+b(x))v=Fv(u,v)+λu.的正基态解的存在性.     

奇异半线性反应扩散方程组Cauchy问题

本文讨论如下问题其中{(б)u/(б)t-(1/tσ)△u=αvp1+β1vp1+f1(x),t>0,x∈RN,(б)u/(б)t-(1/tσ)△v=α2uq2+β2vp2+f2(x),t>0,∈RN,limt→0+u(t,x)=limt→0+v(t,x)=0,x∈Rn,其中σ＞0,pi＞1,qi＞1(i=1,2),α1≥0,α2＞0,β1＞0,β2≥0,fi(x)(i=1,2)连续有界非负,(f1(x),f2(x))(≡／)(0,0).给出了非负局部解存在的几个充分条件和解的爆破结果.     

Existence of ground state solutions to Hamiltonian elliptic system with potentials

In this paper, we investigate nonlinear Hamiltonian elliptic system{-?u + b(向量)(x) · ?u +(V(x) + τ)u = K(x)g(v) in R~N,-?v-(向量)b(x)·?v +(V(x) + τ)v = K(x)f(u) in R~N,u(x) → 0 and v(x) → 0 as |x| →∞,where N ≥ 3, τ 0 is a positive parameter and V, K are nonnegative continuous functions,f and g are both superlinear at 0 with a quasicritical growth at infinity. By establishing a variational setting, the existence of ground state solutions is obtained.     

Liouville type theorems for Schrdinger systems

We study positive solutions to the following higher order Schr¨odinger system with Dirichlet boundary conditions on a half space:(-△)α2 u(x)=uβ1(x)vγ1(x),in Rn+,(-)α2 v(x)=uβ2(x)vγ2(x),in Rn+,u=uxn==α2-1uxnα2-1=0,onRn+,v=vxn==α2-1vxnα2-1=0,onRn+,(0.1)whereαis any even number between 0 and n.This PDE system is closely related to the integral system u(x)=Rn+G(x,y)uβ1(y)vγ1(y)dy,v(x)=Rn+G(x,y)uβ2(y)vγ2(y)dy,(0.2)where G is the corresponding Green’s function on the half space.More precisely,we show that every solution to(0.2)satisfies(0.1),and we believe that the converse is also true.We establish a Liouville type theorem—the non-existence of positive solutions to(0.2)under a very weak condition that u and v are only locally integrable.Some new ideas are involved in the proof,which can be applied to a system of more equations.     

一类带临界指数增长的椭圆型方程组两个正解的存在性

该文主要讨论带临界指数的椭圆型方程组{-Δu + a(x)u =2α/α+βuα-1vβ + f(x),x ∈Ω,-Δv+b(x)v=2β/α+βuαvβ-1+ g(x),x ∈ Ω,(*)u ＞ 0,v ＞ 0,x ∈Ω,u=v=0,x ∈(a)Ω解的存在性,其中Ω是RN中一个光滑有界区域,N=3,4,a≥2,β≥2...     

有限元求解一维奇异摄动问题

1 问题的引入 考虑边值问题 L_y≡-εy″+p(x)y′+q(x)y=f(x),x∈I≡(o,1), y(0)=y(1)=0, (1,1)其中ε是一常数,ε∈(0,1),p(x),q(x),f(x)是[0,1]上的光滑函数,且满足p(x)≥a_1>0,q(x)≥0,q(x)-(1/2)P′(x)≥a_2>0.以下用C和d表示一常数,仅依赖于p(x),q(x),f(x),与ε无关,在不同的地方它们可能代表不同的数. 引入双线性形式 B(u,v)=integral from n=0 to 1(εu′v′+pu′v +quv)dx,u,v∈H~1(I),及范数     

用变换研究曲线间的位置关系

如果令u=mx,v=nx,则这个线性变换将x-O-y平面上的点P(x,y)变为u-O-v平面上的点Q(u,v),将x-O-y平面上的曲线P(x,y)=0变为u-O-v平面上的曲线Q(u,v)=0.这样的线性变换具有如下性质.1)如果P(x,y)=0为直线,那么Q(u,v)=0仍为直线;2)如果P(x,y)=0为曲线,那么Q(u,v)=0仍为曲线,并且变换     

Fujita型反应扩散方程组整体解的存在性、非存在性与渐近性质

本文研究 Ｆｕｊｉｔａ型反应扩散方程组的初值问题：ｕｔ－△ｕ＝ａ１ｕ~α１－１ｕ＋ｂ１ｖ~β１－１ｖ,ｖｔ－△ｖ＝ａ２ｕ~α２－１ｕ＋ｂ２ｖ~β２－１ｖ,ｕ（Ｘ,０）＝ｕ０（Ｘ）,Ｖ（Ｘ,０）＝Ｖ０（Ｘ）,（Ｘ,ｔ）Ｒ~Ｎ ｘ Ｒ~＋,其中 ａｉ,ｂｉ≥ ０, αｉ,βｉ≥ １（ｉ＝ １,２）,给出了非负整体 Ｌ~ｐ解与古典解存在性与非存在性的一系列充分条件,并讨论了解的渐近性质．本文所用方法和所得结果与已有的工作［１－４］,有很大的不同,不但在某些方面推广了［１－５］,而且从某些方面改进了［１］的结果。     

Weak solvability and well-posedness of a coupled Schrödinger-Korteweg de Vries equation for capillary-gravity wave interactions

An interaction equation of the capillary-gravity wave is considered. We show that the Cauchy problem of the coupled Schrödinger-KdV equation,

is locally well-posed for weak initial data . We apply the analogous method for estimating the nonlinear coupling terms developed by Bourgain and refined by Kenig, Ponce, and Vega.

     

Nonexistence Results for the Korteweg–de Vries and Kadomtsev–Petviashvili Equations

We study characteristic Cauchy problems for the Korteweg–de Vries (KdV) equation u_t = uu_x + u_xxx , and the Kadomtsev–Petviashvili (KP) equation u_yy =( u_xxx + uu_x + u_t ) _x with holomorphic initial data possessing non-negative Taylor coefficients around the origin. For the KdV equation with initial value u (0,  x )= u ₀( x ), we show that there is no solution holomorphic in any neighborhood of ( t ,  x )=(0, 0) in C² unless u ₀( x )= a ₀+ a ₁ x . This also furnishes a nonexistence result for a class of y -independent solutions of the KP equation. We extend this to y -dependent cases by considering initial values given at y =0, u ( t ,  x , 0)= u ₀( x ,  t ), u_y ( t ,  x , 0)= u ₁( x ,  t ), where the Taylor coefficients of u ₀ and u ₁ around t =0, x =0 are assumed non-negative. We prove that there is no holomorphic solution around the origin in C³, unless u ₀ and u ₁ are polynomials of degree 2 or lower. MSC 2000: 35Q53, 35B30, 35C10.     

A New Viscous Regularization of the Riemann Problem for Burgers' Equation

This paper gives a new viscous regularization of the Riemann problem for Burgers' equation u_t + (\frac{u²}{2})_z = 0 with Riemann initial data u = u_(x ≤ 0), u = u_+(x > 0} at t = 0. The regularization is given by u_t + (\frac{u²}{2})_z = εe^tu_{zz} with appropriate initial data. The method is different from the classical method, through comparison of three viscous equations of it. Here it is also shown that the difference of the three regularizations approaches zero in appropriate integral norms depending on the data as ε → 0_+ for any given T > 0.     

Non-existence of Global Solutions for a Fractional Wave-diffusion Equations

We considered the Cauchy problem for the fractional wave-diffusion equation $$D^αu-Δ|u|^{m-1}u+(-Δ)^{β/2}D^γ|u|^{l-1}u=h(x,t)|u|^p+f(x,t)$$ with given initial data and where p > 1, 1 < α < 2, 0 < β < 2, 0 < γ < 1. Nonexistence results and necessary conditions for global existence are established by means of the test function method. This results extend previous works.     

关于Fujita型反应扩散方程组的Cauchy问题

本文研究Ｆｕｊｉｔａ型反应扩散方程组ｕｔ－Δｕ＝α１｜ｕ｜ｑ１－１ｕ＋β１｜ｖ｜ｐ１－１ｖ，（ｘ∈ＲＮ，ｔ＞０），ｖｔ－Δｖ＝α２｜ｕ｜ｑ２－１ｕ＋β２｜ｖ｜ｐ２－１ｖ，ｕ（ｘ，０）＝ｕ０（ｘ）０，ｖ（ｘ，０）＝ｖ０（ｘ）０，（ｘ∈ＲＮ）Ｌｐ解的整体存在性和有限时间Ｂｌｏｗ ｕｐ问题．这里ｑｉ＞１，ｐｉ＞１（ｉ＝１，２），α１０，α２＞０，β１＞０，β２０，１ｐ＋∞．     

一类双曲型方程反问题的存在性与唯一性

针对双曲型方程定解问题{utt=a2uxx+f(t),0xπ,a∈R且a≠0,u(0,t)=v1(t),u(π,t)=v2(t),t0,u(x,0)=g(x),ut(x,0)=h(x),0≤x≤π研究了可以唯一决定未知函数组{v1(t),v2(t),f(t)}的基本条件,提出了该定解问题的反问题,并且讨论了此反问题的存在性与唯一性.     

Justification of a Perturbation Approximation of the Klein–Gordon Equation

Consider the nonlinear wave equation
u_tt − γ ² u_xx + f(u) = 0
with the initial conditions
u ( x ,0) = εφ ( x ), u _t( x ,0) = εψ ( x ),
where f ( u ) is either of the form f ( u )= c ² u −σ u ^{2 s +1}, s =1, 2,…, or an odd smooth function with f '(0)>0 and | f '( u )|≤ C ₀².The initial data φ( x )∈ C ² and ψ( x )∈ C ¹ are odd periodic functions that have the same period. We establish the global existence and uniqueness of the solution u ( x ,  t ; ɛ), and prove its boundedness in x ∈ R and t >0 for all sufficiently small ɛ>0. Furthermore, we show that the error between the solution u ( x ,  t ; ɛ) and the leading term approximation obtained by the multiple scale method is of the order ɛ³ uniformly for x ∈ R and 0≤ t ≤ T /ɛ², as long as ɛ is sufficiently small, T being an arbitrary positive number.     

非线性三点共轭边值问题正解存在性的特征值准则

本文提出了三点边值问题-v″(t)=b(t)f(v(t)),满足v′(0)=0及v(1)=αv(η)的共轭问题-u″(t)=b(t)f(u(t)),u′(0)=u(1)=0及u′_+(η)-u′_-(η)=αu′(1),得到了相应的Green函数．将其转化为Hammertein型积分方程,借助于其相应线性问题的第一特征值,利用锥上的不动点指数理论,给出了共轭问题单个正解及多个正解存在的特征值准则．     

一个交通模型的行波解的渐近稳定性

主要考虑下面的交通模型的行波解的渐近稳定性.{vt-ux=0 (E)ut+p(v)x=1/ε(f(v)-u+μuxx 其中初始值为 (v,u)(x,0)=(v0(x),u0(x))→(v±,u±),v±＞0,as x→±∞.(Ⅰ)在允许流函数f不是凹函数以及初始值在无穷远处的极限不满足平衡方程的条件下,我们得到了稳定性...