带有非线性源的以有限Radon测度为初值的拟线性双曲方程的BV解

这篇文章主要考虑下列以有限Radon测度为初值的非线性双曲方程的Cauchy问题u_t+(u~m)_x=u~p,其中m1,1pm是给定常数.特别的,在文中得到了上述问题BV解的存在唯一性.     

以局部有限测度为初值的拟线性退化双曲方程 BV解的存在唯一性

本文讨论了下列拟线性退化双曲方程ut+ψ(u)x=0,以σ-有限的Borel测度为初值条件的Cauchy问题解的存在唯一性,其中ψ在R上是连续的非减的函数.     

以局部有限测度为初值的拟线性退化双曲方程BV解的存在唯一性

本文讨论了下列拟线性退化双曲方程ut (?)(u)x=0,以σ-有限的Borel测度为初值条件的Cauchy问题解的存在唯一性,其中在R上是连续的非减的函数.     

一类非线性双曲型发展方程的孤子解

研究了一类非线性发展方程.首先在无扰动情形下,利用待定函数和泛函同伦映射方法得到了非扰动发展方程的孤子精确解和扰动方程的任意次近似行波孤子解.接着引入一个同伦映射,并选取初始近似函数,再用同伦映射理论,依次求出非线性双曲型发展扰动方程孤子解的各次近似解析解.再利用摄动理论举例说明了用该方法得到的近似解析解的有效性和各次近似解的近似度.最后,简述了用同伦映射方法得到的近似解的意义,指出了用上述方法得到的各次近似解具有便于求解、精度高等优点.     

一类具边界耗散的高阶非线性双曲方程整体解的不存在性

本文利用广义凸性方法证明了边界耗散的非线性四阶方程初边值问题整体解的不存在性     

一类非线性双曲Schrodinger方程的有限差分法

在对深水波的研究中,方程(1.1a)模拟了没有表面张力情况下深水波的包络面(见文[1])。M.J.Ablowitz和H.Segur在[1]中指出:方程(1.1a)不具有Painleve性质,所以很可能逆散射方法对此方程是无效的。在周期边界条件下,方程(1.1a)不存在周期解。但是,数值积分解表明:(1.1a)存在近似周期解。虽然方程(1.1a)与通常的Schrodinger方程仅一符号之差,然而在定解问题的研究上,方程(1.1)较文[3]的方程要难得多,我们希望通过数值解的研究,讨论方程(1.1)之解的性质,诸如周期性、孤立子解的碰撞等。 本文给出了方程(1.1)的“蛙跳”差分格式,采用文[4]的方法,利用有界延拓法,证明了差分解的收敛性与稳定性。最后,利用数值例子,验证了此格式的可信性。     

一类高阶非线性双曲型方程Cauchy问题解的爆破

本文研究一类高阶非线性双曲型方程utt-uxx+μuxxx-αuxxtt+βuxxxxtt=f(ux)x的Cauchy问题,证明问题解的存在性与唯一性,并给出解在有限时刻爆破的充分条件.     

一类非线性双曲型方程初边值问题解的爆破

该文给出一类非线性双曲型方程初边值问题解爆破的充分条件, 并证明问题局部解的存在性与唯一性     

一类高阶非线性双曲型方程初边值问题解的爆破

本文研究一类高阶非线性双曲型方程的初边值问题，证明问题局部广义解的存在性与唯一性，同时给出解爆破的充分条件。     

一类非线性双曲方程整体弱解的存在性与不存在性

讨论了一类非线性双曲方程uu-a1△u-a2n∑i=1 / xi(｜uxi｜^-1)-a3△ut=0的初边值问题整体弱解的存在性与不存在性和指数衰减。     

A Nonlocal Problem with Integral Conditions for the Quasilinear Hyperbolic Equation

In this paper, we consider a nonlocal problem with integral conditions for the quasilinear hyperbolic equation in a rectangular domain. The existence and uniqueness of the generalized solution are established.     

非线性双曲型方程的变网格有限元法

对一类非线性双曲型方程给出了两种变网格有限元逼近格式 .在一定条件下 ,得到了最优 H 1模误差估计     

Cauchy's Problem for Degenerate Quasilinear Hyperbolic Equations with Measures as Initial Values

The aim of this paper is to discuss the Cauchy problem for degenerate quasilinear hyperbolic equations of the form \frac{∂u}{∂t} + \frac{∂u^m}{∂x} = -u^p, m > 1, p > 0 with measures as initial conditions. The existence and uniqueness of solutions are obtained. In particular, we prove the following results: (1) 0 < p < 1 is a necessary and sufficient condition for the above equations to have extinction property; (2) 0 < p < m is a necessary and sufficient condition for the above equations to have localization property of the propagation of perturbations.     

Initial Boundary Value Problem for a Damped Nonlinear Hyperbolic Equation

In the paper, the existence and uniqueness of the generalized global solution and the classical global solution of the initial boundary value problems for the nonlinear hyperbolic equation u_{tt} + k_1u_{xxxx} + k_2u_{xxxxt} + g(u_{xx})_{xx} = f(x, t) are proved by Galerkin method and the sufficient conditions of blow-up of solution in finite time are given.     

Blow-up of Solutions of a Quasilinear Wave Equation with Nonlinear Dissipation

In this paper we establish the blow up of solutions to the quasilinear wave equation with a nonlinear dissipative term u_{tt} - M(||A^{1/2}u||²_2)Au + |ut|^βu_t = |u|^pu x ∈ Ω, t > 0     

Nonexistence of Global Solutions of Some Quasilinear Hyperbolic Equations with Dynamic Boundary Conditions

In this paper the non-existence of global solutions of two fourth-order hyperbolic equations with dynamic boundary conditions is considered. The method of proof makes use of the generalized convexity method due to Ladyzhenskaya and Kalantarov [4].     

二阶拟线性双曲型方程具特征边界的初边值问题

本文讨论问题：其中，方程左端为严格双曲，Σ为特征边界，设右端ｆ（ｔ，ｘ，ｕ，(?)ｕ）在｜ｘ｜充分大时等于０．利用对称双曲组理论及文［５］的结果，通过迭代、模估计，得到了上述问题的Ｘ^＊局部光滑解的存在、唯一性。     

Existence and Uniqueness of BV Solutions for a Class of Degenerate Boltzmann Equations with Measures as Initial Conditions

The existence and uniqueness of the solutions for the Boltzmann equations with measures as initial value are still an open problem which is posed by P. L. Lions （2000）. The aim of this paper is to discuss the Cauchy problem of the system of discrete Boltzmann equations of the form
δtfi＋（fi^mi）=Qi（f1,f2...,fn）,（mi〉1,i-1,...n）
with non-negative finite Radon measures as initial conditions. In particular, the existence and uniqueness of BV solutions for the above problem are obtained.