Burgers方程精确解的一种构造方法

研究了Burgers方程的精确解问题.依据Adomian的分解法，对各种类型的非线性算子，构造出Adomian多项式，给出了Burgers方程的具有初始条件的精确解的求解方法，并利用此方法获得了具体初始条件下的Burgers方程的冲击波解和有理解，同时讨论了解的有关性质.研究工作表明该方法具有相当广泛的适应性.     

Global and blow-up solutions for nonlinear parabolic equations with Robin boundary conditions

In this paper we discuss the blow-up for classical solutions to the following class of parabolic equations with Robin boundary condition: $\{\begin{array}{cc}{\left(b\left(u\right)\right)}_t=??\left(g\left(u\right)?u\right)+f\left(u\right)\hfill & \text{in }\Omega \times (0,T)\text{,}\hfill \\ \frac{?u}{?n}+\gamma u=0\hfill & \text{on }?\Omega \times (0,T)\text{,}\hfill \\ u(x,0)=h\left(x\right)\ge 0\hfill & \text{in }\overline{\Omega }\text{,}\hfill \end{array}$ where $\Omega$ is a bounded domain of ${\mathbb{R}}^N(N\ge 2)$ with smooth boundary $?\Omega$. By constructing some appropriate auxiliary functions and using a first-order differential inequality technique, we derive conditions on the data which guarantee the blow-up or the global existence of the solution. For the blow-up solution, a lower bound on blow-up time is also obtained. Moreover, some examples are presented to illustrate the applications.     

Finite time blow-up for a class of parabolic or pseudo-parabolic equations

In this paper, we study the initial boundary value problem for a class of parabolic or pseudo-parabolic equations:

$u_t?a\Delta u_t?\Delta u+bu=k\left(t\right)|u{|}^{p?2}u,(x,t)\in \Omega \times (0,T),$

where $a\ge 0$, $b>?{?}_1$ with ${?}_1$ being the principal eigenvalue for $?\Delta$ on $H_0^1\left(\Omega \right)$ and $k\left(t\right)>0$. By using the potential well method, Levine’s concavity method and some differential inequality techniques, we obtain the finite time blow-up results provided that the initial energy satisfies three conditions: (i) $J(u_0;0)<0$; (ii) $J(u_0;0)\le d\left(\infty \right)$, where $d\left(\infty \right)$ is a nonnegative constant; (iii) $0<J(u_0;0)\le C\rho \left(0\right)$, where $\rho \left(0\right)$ involves the $L^2$-norm or $H_0^1$-norm of the initial data. We also establish the lower and upper bounds for the blow-up time. In particular, we obtain the existence of certain solutions blowing up in finite time with initial data at the Nehari manifold or at arbitrary energy level.     

Blow-up of solutions for semilinear stochastic delayed reaction–diffusion equations with Lévy noise

This paper is concerned with a class of semilinear stochastic delayed reaction–diffusion equations driven by Lévy noise in a separable Hilbert space. We establish sufficient conditions to ensure the existence of a unique positive solution. Moreover, we study blow-up of solutions in finite time in mean $L^p$-norm sense. Several examples are given to illustrate applications of the theory.     

Breakdown for a Kirchhoff-type Beam with a Fractional Boundary Feedback

A Kirchhoff-type equation describing the transversal vibrations of a beam is considered. The beam is clamped to a rigid base at one part of its edge and free at the remaining part. On the free part, it is subject to a feedback involving fractional derivatives instead of the classical velocity of the deflection and angular velocity. In presence of an external nonlinear source we prove that solutions blow up at a finite time.      

一类具有耗散项的非线性Schrodinger方程的初边值问题解的破裂

本文考虑如下问题这里具有光滑边界，ａ＞０，在适当条件下，得到了问题（＊）的解的破裂定理。     

LLDPE中空容器的研制

研究了以国产LLDPE,HDPE为主要原料的新型中空容器的成型技术,研制的中空容器耐环境应力开裂性、耐刺穿性等物理机械性能均达到或超过SG259-82标准,且重量为普通中空容器的75％～80％,具有推广使用价值。     

一类拟线性热方程的整体解和Blow－up问题

讨论了拟线性热方程Ｃａｕｃｈｙ问题的整体解及Ｂｌｏｗ－ｕｐ问题，给出了有意义的结果。     

Euler-like discrete models of the logistic differential equation

To understand the behavior of difference schemes on nonlinear differential equations, it seems desirable to extend the standard linear stability theory into a nonlinear theory. As a step in that direction, we investigate the stability properties of Euler-related integration algorithms by checking how they preserve and violate the dynamical structure of the logistic differential equation.Among the schemes considered are two linearly implicit nonstandard schemes which are adjoint to each other. We find that these schemes are superior to explicit schemes when they are stable and the blow-up time has not passed: for these λh-values they are dynamically faithful. When these schemes ‘turn unstable’, however, they have much less desirable properties than explicit or fully implicit schemes: they become simultaneously superstable and unstable. This is explained by the fact that these schemes are not self-adjoint: the linearly implicit self-adjoint scheme is dynamically faithful in an Euler-typical range of step sizes and gives correct stability for all step sizes.     

Blow-up phenomena in porous medium equation systems with nonlinear boundary conditions

This paper deals with the blow-up phenomena for the following porous medium equation systems with nonlinear boundary conditions $\left\{\begin{array}{cc}{u}_t=\Delta u^m+k_1\left(t\right)f_1\left(v\right),v_t=\Delta v^n+k_2\left(t\right)f_2\left(u\right)\hfill & in\Omega \times (0,t^1),\hfill \\ \hfill \\ \hfill \\ \frac{?u}{?\nu }=g_1\left(u\right),\frac{?v}{?\nu }=g_2\left(v\right)\hfill & on?\Omega \times (0,t^1),\hfill \\ \hfill \\ \hfill \\ u(x,0)=u_0\left(x\right)\ge 0,v(x,0)=v_0\left(x\right)\ge 0\hfill & in\overline{\Omega },\hfill \end{array}\right.$ where $m,n>1$, $\Omega ?{\mathbb{R}}^N(N\ge 2)$ is bounded convex domain with smooth boundary. Using a differential inequality technique and a Sobolev inequality, we prove that under certain conditions on data, the solution blows up in finite time. We also derive an upper and a lower bound for blow-up time. In addition, as applications of the abstract results obtained in this paper, an example is given.