广义超弹性杆方程关于Neumann问题解的存在性

通过对广义超弹性杆方程的Neumann边界条件及初始条件的研究,得到了广义超弹性杆方程在Neumann边界条件下局部解和整体解的存在.     

PEAKED WAVE SOLUTIONS TO A GENERALIZED HYPERELASTIC-ROD WAVE EQUATION

We study peaked wave solutions of a generalized Hyperelastic-rod wave equation describing waves in compressible hyperelastic-rods by using the bifurcation theory of planar dynamical systems and numerical simulation method. The existence domain of the peaked solitary waves are found. The analytic expressions of peaked solitary wave solutions are obtained. Our numerical simulation and qualitative results are identical.     

一类广义Boussinesq型方程解的爆破

研究一类广义Boussinesq型方程的初边值问题,利用Galerkin方法证明问题局部广义解的存在性与唯一性．同时,通过使用凸性方法给出问题的解在有限时刻发生爆破的充分条件．     

一类非线性退化波动方程解的爆破

考虑了有界区域上一类非线性退化波动方程的初边值问题.通过改进Vitillaro,Li和Tsai的方法,建立了非正的初始能量以及正的初始能量下解的爆破结果.同时,还给出了解的生命跨度估计.     

一类Boussinesq方程解的爆破和不稳定性

研究了一类非线性Boussinesq方程解的爆破和基态解的不稳定性,还得到了问题带在不同域中的初值的整体有界解.     

一类广义KdV方程的孤立波精确解

本文给出求一类广义ＫｄＶ方程的孤立波精确解的方法．     

广义Camassa-Holm方程的行波解

运用平面动力系统理论和方法给出了广义Camassa-Holm方程在各种参数条件下的相图与分支,分析了奇线对其行波解的影响,获得了广义Camassa-Holm方程光滑、非光滑孤立波解和周期波解的存在性及个数,求出了它的两组新周期尖波解的显式表达式.     

广义Lienard方程解的振动性

本文研究了广义Ｌｉｅｎａｒｄ方程解的振动性，在方程具有一个奇点与多个奇点的情形下，给出了使它的所有解振动的一系列充要条件，推广了文［２］，［４］的结果，改进了［５］的结果。     

一类高阶非线性波动方程解的存在性

研究一类高阶非线性波动方程的初边值问题 ,证明问题局部广义解的存在性、唯一性 ,并用凸性方法证明解爆破的充分条件 .     

广义Sawyer-Eliassen方程解的存在性

本文讨论了广义Sawyer-Eliassen方程组的解的存在性问题。     

Blow-up phenomena for the Liouville equation with a singular source of integer multiplicity

We are concerned with the existence of blowing-up solutions to the following boundary value problem

$?\mathrm{\Delta }u=\lambda a(x)e^u?4\pi N{\delta }_0\text{ in }\mathrm{\Omega },u=0\text{ on }?\mathrm{\Omega },$

where Ω is a smooth and bounded domain in ${\mathbb{R}}^2$ such that $0\in \mathrm{\Omega }$, $a(x)$ is a positive smooth function, N is a positive integer and $\lambda >0$ is a small parameter. Here ${\delta }_0$ defines the Dirac measure with pole at 0. We find conditions on the function a and on the domain Ω under which there exists a solution $u_{\lambda }$ blowing up at 0 and satisfying $\lambda {\int }_{\mathrm{\Omega }}a(x)e^{u_{\lambda }}\to 8\pi (N+1)$ as $\lambda \to 0^{+}$.     

Explicit peaked wave solutions to the generalized Camassa-Holm equation

By constructing auxiliary differential equations, we obtain peaked solitary wave solutions of the generalized Camassa-Holm equation, including periodic cusp waves expressed in terms of elliptic functions.     

Bifurcations of traveling wave solutions for a generalized Camassa-Holm equation

In this paper, the traveling wave solutions for a generalized Camassa-Holm equation $u_t-u_{xxt}=\frac{1}{2}(p+1)(p+2)u^pu_x-\frac{1}{2}p(p-1)u^{p-2}u_x^3-2pu^{p-1}u_xu_{xx}-u^pu_{xxx}$ are investigated. By using the bifurcation method of dynamical systems, three major results for this equation are highlighted. First, there are one or two singular straight lines in the two-dimensional system under some different conditions. Second, all the bifurcations of the generalized Camassa-Holm equation are given for $p$ either positive or negative integer. Third, we prove that the corresponding traveling wave system of this equation possesses peakon, smooth solitary wave solution, kink and anti-kink wave solution, and periodic wave solutions.     

The global weak solutions to the Cauchy problem of the generalized Novikov equation

In this paper, we study the Cauchy problem of the generalized Novikov equation. We first show that under suitable condition, the strong solution exists globally via some a priori estimates. Then, we prove the existence and uniqueness of global weak solutions by the approximation method. We also obtain the exact peaked solutions.     

Integral representation for the solution of the stationary Schrödinger equation in a cone

In this paper, we consider the Dirichlet problem for the stationary Schrödinger equation in a cone with continuous boundary data. For a solution u of the stationary Schrödinger equation in a cone, we prove that if its positive part u⁺ satisfying a slowly growing condition, then its negative part u^? can also be dominated by a similar slowly growing condition. Meanwhile, u can be represented by its integral in the boundary of the cone.     

The Liouville equation in a half-plane

We classify the solutions of the equation Δu+aeu=0 in the half-plane that satisfy the Neumann boundary condition ∂u/∂t=ceu/2 on . An analogous problem in the once punctured disc D^∗⊂R² is also solved.     

On the global weak solutions for a modified two‐component Camassa‐Holm equation

In this paper, we investigate the existence of global weak solutions to the Cauchy problem of a modified two‐component Camassa‐Holm equation with the initial data satisfying lim_{x → ±∞}u₀(x) = u_±. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution for the system under the assumption u_? ≤ u₊. The global weak solution is obtained as a limit of approximation solutions. The key elements in our analysis are the Helly theorem and the estimation of energy for approximation solutions in $H^1(\mathbb {R})\times H^1(\mathbb {R})In this paper, we investigate the existence of global weak solutions to the Cauchy problem of a modified two‐component Camassa‐Holm equation with the initial data satisfying lim_{x → ±∞}u₀(x) = u_±. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution for the system under the assumption u_? ≤ u₊. The global weak solution is obtained as a limit of approximation solutions. The key elements in our analysis are the Helly theorem and the estimation of energy for approximation solutions in $H^1(\mathbb {R})\times H^1(\mathbb {R})$ and some a priori estimates on the first‐order derivatives of approximation solutions.     

Peaked wave solutions of CH-r equation

In this paper the qualitative analysis methods of planar autonomous systems and numerica simu-lation are used to investigate the peaked wave solutions of CH-r equation. Some explicit expressions of peakedsolitary wave solutions and peaked periodic wave solutions are obtained,and some of their relationships arerevealed.Why peaked points are generated is discussed.