线性退化的严格双曲组具慢衰减初值的Cauchy问题的经典解

该文给出了线性退化的严格双曲组具慢衰减及小全变差初值的Cauchy问题的经典解的整体存在唯一性. 这个结果进一步推广了A Bressan的相关结果     

出现在双色谱中的非线性双曲型守恒律组整体光滑解的存在性(英文)

本文讨论了出现在双色谱中的非线性双曲型守恒律组的如下Cauchy问题{ut+(u/1+u+v)x=0,vt+(v/1+u+v)x=0,初值为u(x,0)=u0(x),v(x,0)=v0(x)的整体光滑解的存在性和唯一性.分析过程基于对角化方法和特征线法.     

拟线性可约化双曲组经典解的生命区间及其应用

本文对一类拟线性可约化双曲组考察具有紧支集小初值的Cauchy问题: t=0:r=εr_0(x),s=εs_0(x)(2)其中k(v)适当光滑,k(0)>0,并设存在一正整数p≥1使 k’(0)=…k~(p-1)(0)=0 而k~(p)(0)≠0,(3)而r_0(x),s_0(x)为紧支集C~1函数,且不同时为零,证明了其经典解的生命区间T(s)=O(ε~(-p)),并给出了在非线性波动方程中的应用。     

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.     

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.     

用Lebesgue-Stieltjes积分定义的间断解的存在唯一性

在Lebesgue-Stieltjies积分定义的广义解的意义下,证明了一个双曲型方程组Cauchy问题整体广义解的存在性和唯一性.作为经典广义解的自然推广,文中的广义解可以包含某些具奇性的双曲波如δ-波.也适用更一般的方程组.     

一阶非齐次拟线性双曲组的柯西问题整体经典解的存在性

本文考察了弱线性退化的一阶非齐次拟线性严格双曲组具有小初值的柯西问题.在非齐次项满足匹配条件的假设下,给出了精细的波的分解公式,利用这些公式,证明了整体C1解的存在唯一性和稳定性.     

带有非负非线性源的拟线性双曲方程BV解的存在唯一性

主要研究了一类带有非负的非线性源的拟线性双曲守恒律方程的Cauchy问题,其中初值为有限Borel测度.克服了初值和非线性项带来的阻碍,得到了局部BV解的存在唯一性.     

带有一般线性耗散项的p-方程组解的衰减估计

该文主要研究带有一般的线性耗散项的p-方程组Cauchy问题解的衰减性.该文采用Fourier变换的方法,利用基本解方法构造Cauchy问题的解,证明了当初始值有紧支集、耗散项系数满足一定条件时p-方程组Cauchy问题的解任意阶导数具有衰减性.该文的讨论可以推广到更加一般的2×2带耗散项的双曲型方程组的情形.     

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.     

一个早期肿瘤生长模型的数学分析

研究了乳腺癌的早期生长模型（DCIS）,它为耦合了抛物方程、椭圆方程的自由边界问题,运用椭圆型方程的变分理论、抛物方程的L^p理论和压缩映照原理,证明了这个问题局部解的存在惟一性,然后用延拓方法得到了整体解的存在惟一性。     

Global Existence of Classical Solution with Small Initial Total Variation for Quasilinear Linearly Degenerate Hyperbolic Systems

In this paper, the author proves the global existence of classical solution to the Cauchy problem with slowly decaying initial data with small initial total variation for general first order quasilinear linearly degenerate hyperbolic systems. This generalizes the corresponding result of A.Bressan for initial data with compact support.     

Global Solvability in Thermoelasticity with Second Sound on the Semi-axis

In this paper, we consider initial boundary value problem for the equations of one-dimensional nonlinear thermoelasticity with second sound in R^+. First, we derive decay rates for linear systems which, in fact, is a hyperbolic systems with a damping term. Then, using this linear decay rates, we get L^1 and L^∞ decay rates for nonlinear systems. Finally, combining with L^2 estimates and a local existence theorem, we prove a global existence and uniqueness theorem for small smooth data.     

一类非线性耦合双曲型方程组的研究

研究了一类非线性耦合双曲型方程组初边值问题,讨论了初边值条件以及非线性项中指数满足的条件,利用Galerkin方法和紧致性的结果,以及椭圆型方程解的正则性定理,经过严格的推导证明了该问题解的存在唯一性的若干结果.     

Non-linear partial differential equations with discrete state-dependent delays in a metric space

We investigate a class of non-linear partial differential equations with discrete state-dependent delays. The existence and uniqueness of strong solutions for initial functions from a Banach space are proved. To get the well-posed initial value problem we restrict our study to a smaller metric space, construct the dynamical system and prove the existence of a compact global attractor.     

To the question of large-amplitude electron oscillations in a plasma slab

A new approach is proposed for constructing and analyzing piecewise smooth exact solutions of the system of quasilinear hyperbolic equations that models the simplest electron oscillations in a plasma slab. A necessary and sufficient condition for their existence and uniqueness is established. An approximate numerical-analytical solution method is constructed for smooth initial data.     

A problem with nonlocal initial data for one-dimensional hyperbolic equation

In this paper we consider a boundary-value problem for one-dimensional hyperbolic equation with nonlocal initial data in integral form. We prove the existence and uniqueness of the generalized solution.     

Existence and uniqueness for a nonlinear discrete hyperbolic system

The existence, uniqueness and asymptotic behaviour of the solutions to a nonlinear discrete hyperbolic system subject to some extreme conditions and initial data are investigated in a real Hilbert space.     

Existence and uniqueness for a nonlinear parabolic/Hamilton–Jacobi coupled system describing the dynamics of dislocation densities

We study a mathematical model describing the dynamics of dislocation densities in crystals. This model is expressed as a 1D system of a parabolic equation and a first order Hamilton–Jacobi equation that are coupled together. We examine an associated Dirichlet boundary value problem. We prove the existence and uniqueness of a viscosity solution among those assuming a lower-bound on their gradient for all time including the initial time. Moreover, we show the existence of a viscosity solution when we have no such restriction on the initial data. We also state a result of existence and uniqueness of entropy solution for the initial value problem of the system obtained by spatial derivation. The uniqueness of this entropy solution holds in the class of bounded-from-below solutions. In order to prove our results on the bounded domain, we use an “extension and restriction” method, and we exploit a relation between scalar conservation laws and Hamilton–Jacobi equations, mainly to get our gradient estimates.