一类非线性高阶Kirchhoff型方程的初边值问题

本文研究了一类具有非线性耗散项的高阶Kirchhoff型方程的初边值问题.通过构造稳定集讨论了此问题整体解的存在性,应用Nakao的差分不等式建立了解能量的衰减估计.在初始能量为正的条件下,证明了解在有限时间内发生blow-up,并且给出了解的生命区间估计.     

有限初始能量的相对论欧拉方程组光滑解的爆破

在初始资料的某些限制下证明有限初始能量的相对论欧拉方程组柯西问题光滑解的爆破.该文的爆破条件不需要初始资料具有紧支集,部分补充了Pan和Smoller的经典爆破结果(2006).     

任意正初始能量状态下半线性波动方程解的有限时间爆破

研究了具有强阻尼的半线性波动方程的初边值问题.在一定的初值条件下,给出高初始能量状态下解在有限时间爆破的充分条件.另外在任意正初始能量状态下给出解在能量空间中整体存在的充分条件.     

含奇异势和记忆项的四阶抛物方程解的整体存在性与爆破

本文研究一类具有奇异势和记忆项的四阶抛物方程在有界域上的初边值问题.当初值在稳定集中,初始能量在正有界范围内,根据Faedo-Galerkin方法结合Hardy-Sobolev不等式得到了问题解的整体存在性并建立了能量泛函的衰减估计;当初始能量为负时,利用凸方法证明了问题的解在有限时刻爆破并估计了爆破时间上界,该上界依赖于初始能量;当初值位于不稳定集,初始能量有上界时,通过构造辅助泛函获得了一个与初始能量无关的爆破时间上界.     

一类多重非线性抛物方程组解的整体不存在性

本文研究了一类多重非线性抛物方程组初边值问题解的整体不存在性,利用修正的能量扰动法,得到了正初始能量解的整体不存在性结果.     

一类退化半导体方程弱解的唯一性

本文研究了一类退化漂移-扩散模型的混合边值问题.利用能量估计的方法,在边界数据和初始数据是严格正的条件下获得了此问题弱解的唯一性.     

带双调和记忆项的四阶非线性伪抛物方程解的整体存在性和不存在性（英文）

本文致力于带双调和记忆项的四阶非线性伪抛物方程初边值问题的研究.通过应用伽辽金方法、势井理论和相关估计,推导出了整体弱解的存在性.此外,通过应用凹性方法、势井理论和不稳定集的定义,不仅得到了具有非正初始能量(E(0)≤0)的弱解在有限时间爆破的结果,而且得到了具有正初始能量(0E(0)d_θ)的弱解在有限时间爆破的结果.     

具阻尼和源项的非线性粘弹热方程组解的爆破

考虑一个具阻尼和源项的非线性粘弹热方程组的初边值问题.建立一个正初始能量下方程组解的爆破结果.     

具正初始能量的耦合波动方程解的整体不存在

考虑带有非线性阻尼和源项的耦合黏弹性波动方程的初边值问题.假设系统具有正初始能量,在适当的条件下证明了解的整体不存在性定理.     

关于可压液晶方程组的能量有界性

考察可压液晶方程组有关能量的基本性质.证明了只要初始能量有界,此系统的能量就是一个与时间无关的有界量.     

Global existence and blow-up of solutions for a system of Petrovsky equations

The initial-boundary value problem for a system of Petrovsky equations with memory and nonlinear source terms in bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set, and obtain the exponential decay estimate of global solutions. Meanwhile, under suitable conditions on relaxation functions and the positive initial energy as well as non-positive initial energy, it is proved that the solutions blow up in the finite time and the lifespan estimates of solutions are also given.     

Blow-up of solutions for a system of nonlocal singular viscoelastic equations

The main propose of this paper is to study the blow-up of solutions of an initial boundary value problem with a nonlocal boundary condition for a system of nonlinear singular viscoelastic equations. where the blow-up of solutions in finite time with nonpositive initial energy combined with a positive initial energy are shown.     

Solutions of the Boltzmann equation with infinite energy: Existence, stability and long time behavior

This paper is devoted to studying a class of solutions to the nonlinear Boltzmann equation having infinite kinetic energy, these solutions have an upper Maxwellian bound with infinite kinetic energy. Firstly, the existence and stability of this kind of solutions are established near vacuum. Secondly, it is proved that this kind of solutions are stable for any initial data, as a consequence, the Boltzmann equation has at most one solution with infinite kinetic energy. Finally, the long time behavior of the solutions is also established.     

General decay and blow up of solution for a nonlinear wave equation with a fractional boundary damping

The paper deals with the study of global existence of solutions and the general decay in a bounded domain for nonlinear wave equation with fractional derivative boundary condition by using the Lyaponov functional. Furthermore, the blow up of solutions with nonpositive initial energy combined with a positive initial energy is established.     

The vanishing viscosity as a selection principle for the Euler equations: The case of 3D shear flow

We show that for a certain family of initial data, there exist non-unique weak solutions to the 3D incompressible Euler equations satisfying the weak energy inequality, whereas the weak limit of every sequence of Leray–Hopf weak solutions for the Navier–Stokes equations, with the same initial data, and as the viscosity tends to zero, is uniquely determined and equals the shear flow solution of the Euler equations corresponding to this initial data. This simple example suggests that, also in more general situations, the vanishing viscosity limit of the Navier–Stokes equations could serve as a uniqueness criterion for weak solutions of the Euler equations.     

一个模拟趋化现象的广义双曲-抛物系统的光滑解的全局分析

考虑一个模拟趋化现象的广义双曲-抛物系统的Cauchy问题,当动能函数为非线性函数且初始值具有小的L~2能量但其H~2能量可能任意大时,得到了全局光滑解的存在性和渐近行为.这些结果推广了以前的关于动能函数为线性函数或初始值具有小的H~2能量情形下的相关结果,首次获得了关于全局光滑大解方面的结果.这些结果的证明基于构造一个新的非负凸熵和做精细的能量估计.     

On behavior of solutions to a class of nonlinear hyperbolic inverse source problem

This article is concerned with a class of hyperbolic inverse source problem with memory term and nonlinear boundary damping. Under appropriate assumptions on the initial data and parameters in the equation, we establish two results on behavior of solutions. At first we proved stability of solutions when the integral overdetermination tends to zero as time goes to infinity and finally a blow-up result is established for certain solution with positive initial energy.     

A new blow-up condition for semi-linear edge degenerate parabolic equation with singular potentials

In this paper, we consider a class of semi-linear edge degenerate parabolic equation with singular potentials, which was proposed by Chen and Liu [Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equation with singular potentials. Discrete Contin. Dyn. Syst. 2016; 26:661–682.] in which the authors proved the solutions of the model blow up in finite time with low initial energy and critical initial energy. By constructing a new functional, we obtain a new blow-up condition, which demonstrates the possibility of finite time blow-up when the initial energy is larger than the critical initial energy.     

Global nonexistence for logarithmic wave equations with nonlinear damping and distributed delay terms

This paper is concerned with global nonexistence of solutions for a logarithmic wave equation with nonlinear damping and distributed delay terms. Due to the simultaneous presence of nonlinear damping and logarithmic source terms, we have difficulty in use of the concavity method. Applying the energy estimates, we show the global nonexistence of solutions with not only non-positive initial energy but also positive initial energy.     

一类双重退化抛物方程解的存在性及零初始能量下的爆破

本文研究下述双重退化抛物方程的初边值问题:并证明在f满足适当的条件下具零或者负初始能量的解的存在性和在有限时刻的爆破性结果.