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.     

半导体材料科学中的能量输运模型

这是一篇关于半导体材料科学中能量输运模型的综述性章。该在这两个方面讨论了具有绝缘边值的两种典型能量输运模型的初边值问题，一方面得到了一种强解的存在性，另一方面得到了光滑解全局存在性和大时间行为。     

外区域上可压缩Euler方程组的有限时间破裂

本文研究高维(n=2, 3)外区域中可压缩Euler方程组初边值问题解的长时间行为.假定初始密度和速度场在常状态附近有一个紧支集小扰动,本文证明带不可渗透边界条件的初边值问题没有整体解,并进一步给出解关于初值小扰动参数的生命跨度上界估计;引入一种比较简单的试探函数方法,并结合能量恒等式证明主要结论.     

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.     

基于交叉变分的非线性Klein—Gordon方程解的整体存在和爆破

研究非线性Klein-Gordon方程的初边值问题,运用位势井方法,在E(0)d的情况得到了方程解的整体存在和爆破.在临界能量状态得到了整体解的存在性与不存在性.最后使用凸性方法,得到某些具有高初始能量解的爆破.     

具有惯性项和阻尼项的Cahn-Hilliard方程的整体吸引子

本文研究具有惯性项和阻尼项的亚三次非线性Cahn-Hilliard方程的初边值问题.在非线性弱正则的条件下,我们建立弱解的适定性,而不考虑非线性项的一阶导数的下界条件.接着利用弱解的渐近紧和能量解的严格Lyapunov函数的存在性,证明在空间(H~2(?)∩H_0~1(?))×L~2(?)上存在整体吸引子.     

具非线性边界条件的Volterra型泛函微分方程边值问题奇摄动

首先利用微分不等式理论和一些分析技巧,探讨了一类具非线性边界条件的二阶Volterra型泛函微分方程边值问题解的存在性问题.然后通过对右端边界层函数和外部解的构造,进一步研究了一类具小参数的二阶Votterra型非线性边值问题.利用微分中值定理和上、下解方法得到了边值问题解的存在性,并给出了解的关于小参数的一致有效渐近展开式.     

一类具有粘性项的拟线性抛物型方程组*

研究了一类具有非线性源项和粘性项的拟线性抛物型方程组的初边值问题．通过构造稳定集, 证明了此问题整体解的存在性, 并建立了解的长时间行为．同时在放松函数的适当假设条件下, 得到了初始能量非负时解的爆破性质及解的生命区间估计．     

半无界空间上的拟线性抛物型方程解的Blow—up现象

本文研究了拟线性抛物型方程的初边值问题在无界区域D上的全局解存在性问题和局部解的Blow-up问题．利用上、下解方法，并借助Green函数，给出了问题（I）全局解的存在性条件，也给出了局部解发生Blow－up现象的条件     

一类非线性发展方程整体解的存在性、渐近性与解的爆破

研究一类非线性发展方程初边值问题整体弱解的存在性，渐近性和解的爆破问题，证明在关于非线性项的不同条件下，上述初边值问题分别在大初值和小初始能量的情况下存在整体弱解，并且讨论了弱解的渐近性。还证明：在相反的条件下，上述弱解在有限时刻爆破，并且给出了一个实例。     

Immediate regularization of measure-type population densities in a two-dimensional chemotaxis system with signal consumption

This paper deals with the Neumann initial-boundary value problem for a classical chemotaxis system with signal consumption in a disk.In contrast to previous studies which have established a comprehensive theory of global classical solutions for suitably regular nonnegative initial data,the focus in the present work is on the question to which extent initially prescribed singularities can be regularized despite the presence of the nonlinear cross-diffusive interaction.The main result in this paper asserts that at least in the framework of radial solutions immediate regularization occurs under an essentially optimal condition on the initial distribution of the population density.More precisely,it will turn out that for any radially symmetric initial data belonging to the space of regular signed Borel measures for the population density and to L² for the signal density,there exists a classical solution to the Neumann initial-boundary value problem,which is smooth and approaches the given initial data in an appropriate trace sense.     

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.     

General Decay and Blow-Up Results for Nonlinear Fourth-Order Integro-Differential Equation

This study aims at considering an initial-boundary value problem for nonlinear fourth-order viscoelastic equation in a bounded domain. Under suitable conditions of the initial data and of the relaxation function, it is proved that the solution energy is generally decayed. It is also shown that regarding arbitrary positive initial energy, certain solutions blow-up in a finite time.     

ALMOST GLOBAL EXISTENCE OF DIRICHLETINITIAL-BOUNDARY VALUE PROBLEM FORNONLINEAR ELASTODYNAMIC SYSTEMOUTSIDE A STAR-SHAPED DOMAIN

This paper deals with the mixed initial-boundary value problem of Dirichlet type for the nonlinear elastodynamic system outside a star-shaped domain. The almost global existence of solution with small initial data to this problem is proved and a lower bound for the lifespan of solutions is given.     

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO THE SYSTEM OF ONE-DIMENSIONAL NONLINEAR THERMOVISCOELASTICITY

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Global classical solutions to ut − Δ(u2m + 1) + λu = ƒ

The global existence of a classical solution of the initial-boundary value problem or the initial value problem for certain degenerating parabolic equations is established by constructing approximate solutions by the standard Galerkin procedure and applying some differential and integral inequalities when the initial value is smooth enough, has small norm in a suitable sense, and may change sign.     

Blow Up of Solutions to One Dimensional Initial-boundary Value Problems for Semilinear Wave Equations with Variable Coefficients

This paper is devoted to studying the following initial-boundary value problemfor one-dimensional semilinearwave equationswith variable coefficients andwith subcritical exponent: $u_{tt}-∂_x(a(x)∂_xu)=|u|^p, x > 0, t > 0, n=1,$ where $u=u(x,t)$ is a real-valued scalar unknown function in $[0,+∞)×[0,+∞)$, here a(x) is a smooth real-valued function of the variable $x∈(0,+∞)$. The exponents p satisfies $1 < p < +∞$ in (0.1). It is well-known that the number $p_c(1)=+∞$ is the critical exponent of the semilinear wave equation (0.1) in one space dimension (see for e.g., [1]). We will establish a blowup result for the above initial-boundary value problem, it is proved that there can be no global solutions no matter how small the initial data are, and also we give the lifespan estimate of solutions for above problem.     

Well-posedness of the IBVP for 2-D Euler equations with damping

In this paper we focus on the initial-boundary value problem of the 2-D isentropic Euler equations with damping. We prove the global-in-time existence of classical solution to the initial-boundary value problem for small smooth initial data by the method of local existence of solution combined with a priori energy estimates, where the appropriate boundary condition plays an important role.