非线性奇异微分方程边值问题的正解

本文的主要结果为：若f(t,u):(0,1)×(0,+∞→[0,+∞）连续,关于u单调减少,存在实数b＞0使对任意0＜r＜1有（0,1）×（0,∞）,则奇异二阶边值问题有正解的充要条件为,有C1[0,1]正解的充分必要条件为其中α,β,σ,γ是非负实效,且为所述边值问题的Green函数．     

二阶周期边值问题的多重正解

考虑如下周期边值问题:其中x~([1])(t)=p(t)x'(t).总假设p(t)>0,q(t)>0,且f(t,x)是[0,1]×(0,+∞)→[0,+∞)的连续函数,f在z=0可以有奇性.利用锥不动点定理以及格林函数的正性,给出周期边值问题单个和多个正解存在性证明的一种新方法.在实际中,定理的条件很容易验证.     

Global behavior for Whitham type equations on a finite interval

We study the initial-boundary value problem for nonlinear nonlocal equations on a finite interval where λ > 0 and pseudodifferential operator is defined by the inverse Laplace transform. The aim of this paper is to prove the global existence of solutions to the inital-boundary value problem (0.1) and to find the main term of the asymptotic representation in the case of the large initial data.     

THE GLOBAL SOLUTION OF INITIALBOUNDARY VALUE PROBLEM OF HIGHER-ORDER MULTIDIMENSIONAL EQUATION OF CHANGING TYPE

In practical problems there appears the higher-order equations of changing type. But,there is only a few of papers, which studied the problems for this kind of equations. In this paper a kind of the higher-order m     

WEAK TRAVELLING WAVE FRONT SOLUTIONS OF GENERALIZED DIFFUSION EQUATIONS WITH REACTION

The author demonstrate that the two-point boundary value problem {p′(s)=f′(s)-λp^β(s)for s∈(0,1);β∈(0,1),p(0)=p(1)=0,p(s)>0 if s∈(0,1),has a solution(λ^-，p^-（s）),where |λ^-| is the smallest parameter,under the minimal stringent restrictions on f(s), by applying the shooting and regularization methods. In a classic paper, Kohmogorov et.al.studied in 1937 a problem which can be converted into a special case of the above problem. The author also use the solution(λ^-,p^-(s)) to construct a weak travelling wave front solution u(x,t)=y(ξ),ξ=x-Ct,C=λ^-N/(N+1),of the generalized diffusion equation with reaction δ/δx（k（u）|δu/δx|^n-1 δu/δx)-δu/δt=g（u），where N>0,k(s)>0 a.e.on(0,1),and f(a):=n+1/N∫0ag(t)k^1/N(t)dt is absolutely continuous ou[0,1],while y(ξ) is increasing and absolutely continuous on (-∞,+∞) and (k(y(ξ))|y′(ξ)|^N)′=g(y(ξ))-Cy′(ξ)a.e.on(-∞，+∞）,y(-∞)=0,y(+∞)=1.     

Hölder continuity of local minimizers of vectorial integral functionals

We study the regularity of vector-valued local minimizers in $ W^{1,p}, p > 1 $, of the integral functional where is an open set in $ \mathbb{R}^N $ and f is a continuous function, convex with respect to the last variable, such that $ 0 \leq f(x,u,t)\leq C(1+t^p) $.We prove that if f = f(x, t), or f = f(x, u, t) and $ p \leq N $, then local minimizers are locally Hölder continuous for any exponent less than 1. If f = f(x, u, t) and p < N then local minimizers are Höolder continuous for every exponent less than 1 in an open set $ \Omega_0 $ such that the Hausdorff dimension of $ \Omega \backslash \Omega_0 $ is less than N–p.AMS Subject Classification: 49N60.     

一维非齐次BBM方程周期边界问题的整体吸引子

研究了一维非齐次方程BBM方程ut-uxxt-αφ(u)x=g(x)+βf(u)+γuxx(α>0,β>0,γ>0),u(x+2π,t)=u(x,t),u(x,0)=u0(x)的周期边界问题.利用Sobolev插值不等式,对解做关于时间t的一致性先验估计,证明了该问题的整体吸引子的存在性.     

非线性四阶周期边值问题多个正解的存在性

利用不动点和度理论,证明了四阶周期边值问题u(4)(t)-βu″(t)+αu(t)=λf(t,u(t)),0≤t≤1,u(i)(0)=u(i)(1),i=0,1,2,3,至少存在两个正解,其中β>-2π2,0<α<(1/2β+2π2)2,α/π4+β/π2+1>0,f:[0,1]×[0,+∞)→[0,+∞)是连续函数,λ>0是常数.     

AN IMPROVEMENT OF A RESULT OF IVOCHKINA AND LADYZHENSKAYA ON A TYPE OF PARABOLIC MONGE-AMP\`ERE EQUATION

For the initial-hotmdary value problem about a type of parabolic Monge-Ampere equation of the form (IBVP): {-Dtu + (deD^2xu)^1/n = f(x,t), (x,t) ∈ Q = Ω&#215;(0,T)}, u(x,t) =Ф(x,t)(x,t) ∈δpQ}, where Ω is a bounded convex domain in R^n, the result in [4] by Ivochkina and Ladyzheokaya is improved in the sense that, under assumptions that the data of the problempossess lower regularity and satisfy lower order compatibility conditions than than in [4], the existence of classical solution to (IBVP) is still established (see Theorem 1.1 below). This cannot be reallzed by only using the method in [4]. The main additional effort the authors have done is a kind of nonlinear perturbation.     

ON THE BOUNDED AND UNBOUNDED SOLUTIONS OF ONE DIMENSIONAL NONLINEAR REACTION－DIFFUSION PROBLEM

ＯＮＴＨＥＢＯＵＮＤＥＤＡＮＤＵＮＢＯＵＮＤＥＤＳＯＬＵＴＩＯＮＳＯＦＯＮＥＤＩＭＥＮＳＩＯＮＡＬＮＯＮＬＩＮＥＡＲＲＥＡＣＴＩＯＮ－ＤＩＦＦＵＳＩＯＮＰＲＯＢＬＥＭ￥ＧＥＷＥＩＧＡＯＲ．Ｏ．ＷＥＢＥＲＡｂｓｔｒａｃｔ：Ｔｈｅｅｘｉｓｔｅｎｃｅｏｆｂ...     

The Chebyshev Spectral Method for Burgers-Like Equations

The Chebyshev polynomials have good approximation properties which are not affected by boundary values. They have higher resolution near the boundary than in the interior and are suitable for problems in which the solution changes rapidly near the boundary. Also, they can be calculated by FFT. Thus they are used mostly for initial-boundary value problems for P.D.E.'s (see [1, 3-4, 6, 8-11]). Maday and Quarterom discussed the convergence of Legendre and Chebyshev spectral approximations to the steady Burgers equation. In this paper we consider Burgers-like equations.$$\begin{cases}∂_iu+F(u)_x-vu_{zx}=0, & -1≤x≤1, 0＜t≤T \\ u (-1,t) =u (1,t) =0, & 0≤t≤T & (0.1)\\ u (x,0) =u_0(x), & -1≤x≤1\end{cases}$$ where $F\in C(R)$ and there exists a positive function $A\in C(R)$ and a constant $p＞1$ such that $$|F(z+y)-F(z)|\leq A(z)(|y|+|y|^p).$$ We develop a Chebyshev spectral scheme and a pseudospectral scheme for solving (0.1) and establish their generalized stability and convergence.     

Stabilization of solutions of initial-boundary problems for quasilinear parabolic equations

Lower bounds are obtained for solutions of the initial-boundary Dirichlet problem for high order equations. Sharp bounds are also obtained for ess sup¦u(x, t)¦ of the Neumann initial-boundary problem for a second- order equation in D=x(t >0), where (Rⁿ, n 2 is a domain with noncompact convex boundary.Translated from Ukrainskii Maternaticheskii Zhurnal, Vol. 44, No. 10, pp. 1441–1450, October, 1992.     

Blow-up vs. Global Finiteness for an Evolution $p$-Laplace System with Nonlinear Boundary Conditions

In this paper, the authors consider the positive solutions of the system of the evolution $p$-Laplacian equations $$\begin{cases} u_t ={\rmdiv}(| ∇u |^{p−2} ∇u) + f(u, v), & (x, t) ∈ Ω × (0, T ), & \\ v_t = {\rmdiv}(| ∇v |^{p−2} ∇v) + g(u, v), &(x, t) ∈ Ω × (0, T) \end{cases}$$with nonlinear boundary conditions $$\frac{∂u}{∂η}= h(u, v), \frac{∂v}{∂η} = s(u, v),$$and the initial data $(u_0, v_0)$, where $Ω$ is a bounded domain in$\boldsymbol{R}^n$with smooth boundary $∂Ω, p > 2$, $h(· , ·)$ and $s(· , ·)$ are positive $C^1$ functions, nondecreasing in each variable. The authors find conditions on the functions $f, g, h, s$ that prove the global existence or finite time blow-up of positive solutions for every $(u_0, v_0)$.     

A global approach to fully nonlinear parabolic problems

We consider the general initial-boundary value problem

(1)        
(2)        
(3)        
where is a bounded open set in with sufficiently smooth boundary.  The problem (1)-(3) is first reduced to the analogous problem in the space with zero initial condition and

The resulting problem is then reduced to the problem where the operator satisfies Condition  This reduction is based on a priori estimates which are developed herein for linear parabolic operators with coefficients in Sobolev spaces.  The local and global solvability of the operator equation are achieved via topological methods developed by I. V. Skrypnik. Further applications are also given involving relevant coercive problems, as well as Galerkin approximations.

     

INVARIANT SETS AND THE HUKUHARA-KNESER PROPERTY FOR PARABOLIC SYSTEMS

In this paper we are concerned with the nonlinear boundary value problem forparabolic system(Lu=f(x,t,u,▽u),x∈Ω,0相似文献   

变系数四阶边值问题正解存在性

该文结合算子谱论,应用锥不动点定理,建立了四阶边值问题\[\left\{ {\begin{array}{l}u^{(4)} + B(t){u}' - A(t)u = f(t,u),0 < t < 1 ,\\u(0) = u(1) = {u}'(0) = {u}'(1) = 0 \end{array}} \right.\]正解存在性定理,这里$A(t),B(t) \in C[0,1]$,$f(t,u):[0,1]\times[0,\infty ) \to [0,\infty )$连续.     

一类椭圆问题正解的存在性和唯一性

该文讨论了二阶拟线性椭圆型问题u|\-\{Ω=0: -div［(d+|u|\+2)\+\{〖SX(〗p〖〗2〖SX)〗-1u］ =λ\-1u\+\{p-1+g(x,u),〓 x∈Ω正解的存在性和唯一性，其中 Ω是 R\+N 中的有界区域, λ\-1 是-△\-p 在 Ω上对应于零Dirichlet边界条件的第一特征根, g(x, t) 满足增长条件lim[DD(X]t→+∞[DD)]〖SX(〗g(x,t)〖〗t\+\{p-1〖SX)〗=0, p>1, 0≤d<+∞〖HT5”H〗关键词:〖HT5”SS〗拟线性椭圆问题； 鞍点； 正解.     

一类四阶常微分方程周期边值问题的正解

本文研究了四阶周期边值问题{u⁴(t)-βu″(t)+αu(t)=f(t,u(t),u′(t),u″(t),u′′′(t)),t∈[0,1],uⁱ(0)=uⁱ(1),i=0,1,2,3正解的存在性,其中f:[0,1]×[0,+∞)×R³→[0,+∞)连续.利用锥上的不动点指数理论,获得了该问题正解的存在性结果,推广了已有文献的相关结果.     

Extinction of Weak Solutions for Nonlinear Parabolic Equations with Nonstandard Growth Conditions

This paper deals with the extinction of weak solutions of the initial and boundary value problem for $u_t$ = div$((|u|^σ + d_0)|∇u|^{p(x)−2}∇u)$. When the exponent belongs to different intervals, the solution has different singularity (vanishing in finite time).     

二阶三点边值问题的正解

该文讨论了二阶三点边值问题$-u'(t)=b(t)f(u(t))$满足$u'(0)=0$, $u(1)={\alpha}u({\eta})$ 正解的存在性与多重性, 其中常数$\alpha, \eta\in(0,1)$, $f\in C ([0,\infty),[0,\infty) )$, $b\in C ([0,1],[0,\infty) )$且存在$t_0\in[0,1]$使$b(t_0)>0$. 利用该问题相应的Green函数, 将其转化为Hammerstein型积分方程, 借助于锥上的不动点指数理论,给出了该问题单个正解和多个正解存在的与其相应线性问题的第一特征值有关的最佳充分性条件.