On solutions of second and fourth order elliptic equations with power-type nonlinearities

We study second and fourth order semilinear elliptic equations with a power-type nonlinearity depending on a power p$p$ and a parameter λ>0$\lambda >0$. For both equations we consider Dirichlet boundary conditions in the unit ball B⊂Rⁿ$B\subset {\mathbb{R}}^n$. Regularity of solutions strictly depends on the power p$p$ and the parameter λ$\lambda$. We are particularly interested in the radial solutions of these two problems and many of our proofs are based on an ordinary differential equation approach.     

带有幂型的广义Zakharov方程组解的爆破率的下界估计

主要研究了关于R~2中一类带有幂型非线性的广义Zakharov方程组的Cauchy问题的有限时间爆破解的爆破率的下界估计.在α≤0和p≥3条件下,对于Cauchy问题任意给定的属于能量空间H~1(R~2)×L~2(R~2)×L~2(R~2)的有限时间的爆破解,得到了对于t靠近有限爆破时间T时的爆破率的最优下界估计.此外,给出了Cauchy问题维里等式的一个应用.     

三维Navier-Stokes方程组关于速度梯度两个分量的正则性准则的一个改进

该文考虑三维Navier-Stokes方程组的Cauchy问题,得到了一个改进的、各向异性的、关于速度梯度的两个分量的正则性准则.此准则改进了文献[24]中的结果.     

带Hardy位势的双调和椭圆方程的正负解及变号解

本文讨论带Hardy位势的四阶渐近线性椭圆方程,应用变分方法,我们得到了正负解及变号解的存在性.     

Uniqueness of the Weak Extremal Solution to Biharmonic Equation with Logarithmically Convex Nonlinearities

In this note, we investigate the existence of the minimal solution and the uniqueness of the weak extremal （probably singular） solution to the biharmonic equation
△2ω=λg（ω）
with Dirichlet boundary condition in the unit ball in Rn, where the source term is logarithmically convex. An example is also given to illustrate that the logarithmical convexity is not a necessary condition to ensure the uniqueness of the extremal solution.     

High Accuracy Solution of Three-Dimensional Biharmonic Equations

In this paper, we consider several finite-difference approximations for the three-dimensional biharmonic equation. A symbolic algebra package is utilized to derive a family of finite-difference approximations for the biharmonic equation on a 27 point compact stencil. The unknown solution and its first derivatives are carried as unknowns at selected grid points. This formulation allows us to incorporate the Dirichlet boundary conditions automatically and there is no need to define special formulas near the boundaries, as is the case with the standard discretizations of biharmonic equations. We exhibit the standard second-order, finite-difference approximation that requires 25 grid points. We also exhibit two compact formulations of the 3D biharmonic equations; these compact formulas are defined on a 27 point cubic grid. The fourth-order approximations are used to solve a set of test problems and produce high accuracy numerical solutions. The system of linear equations is solved using a variety of iterative methods. We employ multigrid and preconditioned Krylov iterative methods to solve the system of equations. Test results from two test problems are reported. In these experiments, the multigrid method gives excellent results. The multigrid preconditioning also gives good results using Krylov methods.     

Rn上带奇异性的非线性双调和方程的正整解

该文以Schauder-Tychonoff不动点定理为工具,建立了一类Rⁿ上带奇异性的非线性双调和方程 Δ²u=f(|x|, u,| u|)u^-β (x ∈ Rⁿ, n ≥ 3, β > 0) 正的径向对称整体解的存在性定理,并给出了解的有关性质,所得的结果丰富和发展了文献[1--5]的结果.     

对带混合非线性项的基尔霍夫方程规范解问题的补充

在本文中,我们研究了带有质量约束条件的基尔霍夫方程规范解的存在性问题, $$-\Big(a+b \int_{\mathbb{R}^{3}}|\nabla u|^{2} \text{d} x\Big)\Delta u=\lambda u+\mu|u|^{q-2} u+|u|^{p-2} u,~~x\in \mathbb{R}^{3},$$ 其中质量约束条件为$$S_{c}:=\Big\{u \in H^{1}(\mathbb{R}^{3}):\int_{\mathbb{R}^{3}}|u|^{2} \text{d} x=c\Big\},$$ 这里$a$, $b$, $c>0$, $\mu\in \mathbb{R}$, $20$, 当$(p, q)$ 属于$\mathbb{R}^{2}$中的某个域时. 我们通过使用约束最小化、集中紧性原理和Minimax方法证明了规范解的存在性.我们部分地扩展了已经被研究的结果.     

Fourier-Galerkin Method for Localized Solutions of Equations with Cubic Nonlinearity

Using a complete orthonormal system of functions in L ²(– ,) a Fourier-Galerkin spectral technique is developed for computing of the localized solutions of equations with cubic nonlinearity. A formula expressing the triple product into series in the system is derived. Iterative algorithm implementing the spectral method is developed and tested on the soliton problem for the cubic Boussinesq equation. Solution is obtained and shown to compare quantitatively very well to the known analytical one. The issues of convergence rate and truncation error are discussed.     

具不对称非线性项的二阶常微分方程周期解的存在性

本文利用变分方法给出了具不对称非线性项的二阶常微分方程周期解的存在性的充分条件,通过例子说明本文的主要结果可看作是Landesman-Lazer条件的补充。     

四阶具强阻尼非线性波动方程解的整体存在性与不存在性

研究四阶具强阻尼非线性波动方程utt-△u+△~2u-α△ut=f(u)的初边值问题.利用位势井方法,得到了问题整体解存在性与不存在性的最佳条件(门槛结果).     

A Counterexample to C 2,1 Regularity for Parabolic Fully Nonlinear Equations

We address the self-similar solvability of a singular parabolic problem and show that solutions to parabolic fully nonlinear equations are not expected to be C ^2,1.     

On the maximum and antimaximum principles for the beam equation

We consider a one-dimensional beam equation with a strictly positive forcing term. Under different assumptions on the restoring force we prove the existence of positive and negative solutions, respectively. Our method is based on non-monotone but convergent iterations and extends related results known in the literature so far.     

含间断项的非线性积分、微分方程极解的存在性

在不涉及紧型条件和耗散型条件的情形下，证明了Ｂａｎａｃｈ空间中含间断项的非线性积分、微分方程极解的存在性，推广了文［１］、［２］和［３］中的结果．     

Elliptic Equations with Degenerate Coercivity： Gradient Regularity

In this paper, we prove higher integrability results for the gradient of the solutions of some elliptic equations with degenerate coercivity whose prototype is where for example, a(x,u)=(1+|u|)^−θ with θ ∈ (0,1). We study the same problem for minima of functionals closely related to the previous equation.     

Structure of Positive Solutions of Quasilinear Elliptic Equations with Critical and Supercritical Growth

The structure of positive radial solutions to a class of quasilinear elliptic equations with critical and supercritical growth is precisely studied. A large solution and a small solution are obtained for the equations. It is shown that the large solution is unique, its asymptotic behaviour and flat core are also discussed.     

Extremal Solutions of a Class of Nonlinear Integro-Differential Equations in Banach spaces

The monotone iterative technique is applied to a class of nonlinear first order integro-differential equations in Banach spaces. First a linear system with a ``small' nonlinear perturbation is solved using Banach's Contraction Principle and the technique of Green's function. Then based upon a comparison result, the existence of minimal and maximal solutions is proved.

     

Stability of Stochastic Delay Evolution Equations with Monotone Nonlinearity

In this paper, we consider the exponentially asymptotic stability of the mild solutions of semilinear stochastic delay evolution equations of monotone type. An Itô-type inequality is our main tool to study the stability in the p-th moment and almost sure sample-path stability of the mild solutions. At last, we give some examples to illustrate the applications of the theorems.