四阶差分方程边值问题解的多重性

利用一个临界点存在性定理,结合上下解方法,证明了一类非线性四阶差分方程边值问题至少有四个解.     

多维Landau-Lifshitz方程的δ-黏性解

本文引入δ-黏性上解,δ-黏性下解和δ-黏性解的概念,给出一些相关性质,利用这些性质证明取值于三维单位球面的多维Landau-Lifshitz方程的δ-黏性上解和δ-黏性下解的存在性,揭示存在两个不相交的开子集M和N,使得δ-黏性上解和δ-黏性下解在M内任一紧子集上趋于(0,1,0),在N内任一紧子集上趋于(0,-1,0)．     

Periodic traveling curved fronts in reaction-diffusion equation with bistable time-periodic nonlinearity

This paper is concerned with the existence and stability of periodic traveling curved fronts for reaction-diffusion equations with time-periodic bistable nonlinearity in two-dimensional space. By constructing supersolution and subsolution, we prove the existence of periodic traveling wave fronts. Furthermore, we show that the front is globally stable.     

Existence and global stability of traveling curved fronts in the Allen-Cahn equations

This paper is concerned with existence and stability of traveling curved fronts for the Allen-Cahn equation in the two-dimensional space. By using the supersolution and the subsolution, we construct a traveling curved front, and show that it is the unique traveling wave solution between them. Our supersolution can be taken arbitrarily large, which implies some global asymptotic stability for the traveling curved front.     

A quasilinearization method for elliptic problems with a nonlinear boundary condition

We study a nonlinear elliptic second order problem with a nonlinear boundary condition. Assuming the existence of an ordered couple of a supersolution and a subsolution, we develop a quasilinearization method in order to construct an iterative scheme that converges to a solution. Furthermore, under an extra assumption we prove that the convergence is quadratic.     

Further results of existence of positive solutions of elliptic Kirchhoff equation with general nonlinearity of source terms

The paper deals with some existence results for elliptic Kirchhoff equations with respect to general nonlinear source terms and changing sign data, where we have used three different methods: direct variational method, Galerkin approach, and subsolution–supersolution method.     

几类具Pucci算子的椭圆方程组解的存在性

研究完全非线性椭圆方程组解的存在性问题,其中ΩR~n,n≥2是有界光滑区域,—Μ_(λ,Λ)~+为具参数0<λ≤Λ的Pucci算子.首先,对f_i,i=1,2为一致有界函数的情形,证明了此方程组存在有界非负解.其次,当{f_1,f_2}是拟增的,且方程组存在有序上、下解时,利用上、下解方法,并结合增算子的不动点定理证明了此方程组存在最大非负解和最小非负解.当{f_1,f_2}是拟减或混拟单调时,使用Schauder不动点定理证明了此方程组至少存在一个非负解.针对此方程组中f_i,i=1,2的某些特殊形式,证明了相应方程组正解的存在性.最后给出了应用实例.     

Critical curves for fast diffusive polytropic filtration equations coupled through boundary

This article deals with the critical curve of a fast diffusive polytropic filtration system coupled at the boundary condition. By constructing the self-similar supersolution and subsolution, we obtain the critical global existence curve. The critical curve of Fujita type is conjectured with the aid of some new results.     

Algebraic criteria for global existence or blow-up for a boundary coupled system of nonlinear diffusion equations

This article deals with the critical curves for a degenerate parabolic system coupled via nonlinear boundary flux. By constructing the self-similar supersolution and subsolution, we obtain the critical global existence curve. The critical curve of Fujita type is conjectured with the aid of some new results.     

抽象二阶周期边值问题解的存在性

研究了有序Banach空间中二阶周期边值问题解的存在性,利用凸锥理论与上、下解方法,获得了解的存在性结果.     

一个反应方程的解与blow—up问题

本文用上、下解的方法讨论了一个推广的核反应数学模型的解的存在性及在有限时刻爆炸（ｂｌｏｗ－ｕｐ）的问题．     

Global existence and nonexistence for a doubly degenerate parabolic system coupled via nonlinear boundary flux

This article deals with the degenerate parabolic system with nonlinear boundary flux. By constructing the self-similar supersolution and subsolution, we obtain the critical global existence curve and the critical Fujita curve for the problem. Especially for the blow-up case, it is rather technical. It comes from the construction of the so-called Zel’dovich-Kompaneetz-Barenblatt profile.     

一类非线性抛物方程组解的存在性和Blow—up

本文应用上下解方法,给出一类非线性抛物方程组整体解存在的充分必要条件的新结果,并且讨论了其解的Blow-up。     

Critical curves for fast diffusive non-Newtonian equations coupled via nonlinear boundary flux

This paper deals with the critical curve for a nonlinear boundary value problem of a fast diffusive non-Newtonian system. We first obtain the critical global existence curve by constructing the self-similar supersolution and subsolution. And then the critical Fujita curve is conjectured with the aid of some new results.     

POSITIVE EQUILIBRIUM SOLUTIONS OF SEMILINEAR PARABOLIC EQUATIONS

The author studies semilinear parabolic equations with initial and periodic boundary value conditions. In the presence of non-well-ordered sub- and super-solutions: "subsolution (?) supersolution",the existence and stability/instability of equilibrium solutions are obtained.     

Multiple solutions of some fourth-order boundary value problems

For some fourth-order boundary value problems, several new existence theorems on multiple positive, negative and sign-changing solutions are obtained. The critical point theory and the supersolution and subsolution method are employed to discuss this problem.     

关于离散HJB方程的下解与上解

本文讨论HJB方程上、下解的几个性质,并讨论了从上解出发的一个迭代法的收敛性.     

一类非线性二阶椭圆型方程组的正解

本文讨论二阶非线性椭圆边值问题的正解的存在性,其中非线性项f和g关于u,v的增长限制很不相同.f是超线性的,而g满足次线性的条件.利用拓扑度理论和上、下解方法,得到了几个正解的存在性定理.作为应用,本文还给出了一些具体的例子.     

Banach空间半线性发展方程的周期解

把上、下解方法引入到有序Ｂａｎａｃｈ空间中的半线性发展方程周期解问题，利用正算子半群特征与单调迭代程序，获得了最大周期解与最小周期解的存在性．所得的结果概括和推广了常微分方程与偏微分方程中的有关结论     

Perron's method for second order hyperbolic PDE. Part I

In parts I, II, and III combined of this paper, we define a notion of viscosity solution for these equations and existence is proved by a Perron-like method. Here, in part I, we prove useful identities, and a maximum-like principle for smooth sub(super) solutions of the standard wave equation. We define a new potential theoretic (P) notion of solution, subsolution and supersolution, and a related potential type (P) Cauchy problem for semilinear second order hyperbolic equations.