一类拟线性波方程的整体光滑解

In this paper, we study the Cauchy problem for the following quasi-linear wave equation utt-2kuxxtt=β(uxn)x, where k>0 and βare real numbers, and n≥2 is an integer. We prove that for any T>0, the Cauchy problem admits a unique global smooth solution u ∈C∞((0, T); H∞(R))∩C ([0, T]; H2(R))∩C1([0, T]; L2(R)) under suitable assumptions on the initial data.     

一类三阶拟线性发展方程的整体解

本文研究一类三阶拟线性发展方程ｕｔｔ－Δｕｔ＋ｕｔ＝ｎｉ＝１ｘｉσｉ（ｕｘｉ），（ｘ，ｔ）∈Ω×（０，Ｔ）的初边值问题，其中ΩＲｎ（ｎ≥１）为一有界域．证明了只要σｉ（ｓ）∈Ｃ１，σ′ｉ（ｓ）（ｉ＝１，…，ｎ）有界并且初始函数满足一定的条件，则上述问题存在唯一的整体弱解．     

欧拉 - 玻尔兹曼方程整体光滑解

本文证明了R^d 中具有某一类小初值的等熵欧拉 - 玻尔兹曼方程整体光滑解的存在性.本文首先构造了等熵欧拉 - 玻尔兹曼方程的局部解, 并证明了局部解的适定性. 此外,文中还构造了关于原方程的随时间 t 增加、具有良好的衰减性质的整体光滑背景解. 同时, 当方程的辐射项系数满足一定条件时, 本文建立了关于源项的估计.通过将背景解的衰减与源项的估计结合起来, 文中证明了存在整数 s>d/2 + 1 ,使得背景解与原方程解的 H^s(R^d)x L²(R₊ x S^d-1;H^s(R^d))范数之差始终是有界的, 从而保证了原方程整体光滑解的存在性.     

三维可压等熵Euler方程光滑解的整体存在性

研究了三维可压等熵Euler方程Cauchy问题光滑解的整体存在性.如果初值是一个常状态的小扰动并且初速度的旋度等于零,证明了三维可压等熵Euler方程Cauchy问题光滑解的整体存在性.     

一类拟线性退缩抛物型方程整体解的存在性与衰减估计

本文讨论了一类拟线性抛物型方程初边值问题整体解的存在性和衰减估计.所得结果改进并推广了文献[1]的相应结果.     

一类半线性积分微分方程初值问题之整体解的存在性与唯一性

本文运用能量廷拓方法,讨论了一类描述有内部热耗散的杆中热传导问题的半线性积分微分方程初值问题之整体古典解的存在性,并附带证明了该问题之古典解的唯一性。     

某类拟线性Burger型方程的波前解的稳定性

本文研究了一些拟线性Burgers型方程的波前解的存在性、稳定性，利用谱分析的方法，证明了光滑波前解在某些加权空间中的渐近稳定性。     

一类拟线性椭圆型方程下解的先验估计

本文在一定的结构条件下,对如下形状的拟线性椭圆型方程∫Ｇｖ·Ａ（ｘ,ｕ,ｕ）＋ｖＢ（ｘ,ｕ,ｕ）｛｝ｄｘ＝０,ｖ∈Ｗ１ｐ（Ｇ）的下解作出了新的先验估计,发展了Ｇｉｌｂａｒｇ＿Ｔｒｕｄｉｎｇｅｒ的某些结果·     

一阶拟线性偏微分方程Cauchy问题的整体光滑解

本文应用整体隐函数存在定理研究了一阶拟线性偏微分方程及主部相同的方程组的整体光滑解问题。     

Almost Global Strong Solutions to Quasilinear Dissipative Evolution Equations

The author proves a global existence result for strong solutions to the quasilinear dissipative hyperbolic equation （1.1） below, corresponding to initial values and source terms of arbitrary size, provided that the hyperbolicity parameter ε is sufficiently small. This implies a corresponding global existence result for the reduced quasilinear parabolic equation （1.4） below.     

Nonresonance and Global Existence of Nonlinear Elastic Wave Equations with External Forces

This paper establishes the global existence of classical solution to the system of homogeneous,isotropic hyperelasticity with time-independent external force,provided that the nonlinear term obeys a ty...     

The Blowup Mechanism of Small Data Solutions for the Quasilinear Wave Equations in Three Space Dimensions

For a class of three-dimensional quasilinear wave equations with small initial data, we give a complete asymptotic expansion of the lifespan of classical solutions, that is, we solve a conjecture posed by John and Hörmander. As an application of our result, we show that the solution of three-dimensional isentropic compressible Euler equations with irrotational initial data which are a small perturbation from a constant state will develop singularity in the first-order derivatives in finite time while the solution itself is continuous. Furthermore, for this special case, we also solve a conjecture of Alinhac.     

Global Existence for Some 4-D Quasilinear Wave Equations with Low Regularity

In this paper, we prove the global existence for some 4-D quasilinear wave equations with small, radial data in H~3× H~2. The main idea is to exploit local energy estimates with variable coefficients, together with the trace estimates.     

Small data solutions of 2-D quasilinear wave equations under null conditions

2 Abstract For the 2-D quasilinear wave equation(?_t~2-?_x)u+2∑ij=0g~(ij)(?u)?_(ij)u = 0 satisfying i,j=0 null condition or both null conditions, a blowup or global existence result has been shown by Alinhac. In this paper, we consider a more general 2-D quasilinear wave equation(?_t~2-?_x)_u+2∑ij=0g~(ij)(?u)?_(ij)u = 0 satisfying null conditions with small initial data and the coefficients i,j=0 depending simultaneously on u and ?u. Through construction of an approximate solution,combined with weighted energy integral method, a quasi-global or global existence solution are established by continuous induction.     

一类含测度的非强制拟线性椭圆方程解的存在性和不存在性

本文主要研究如下含非线性梯度项的非强制拟线性椭圆方程\begin{equation*}\left \{\begin{array}{rl}-\text{div}(\frac{|\nabla u|^{p-2}\nabla u}{(1+|u|)^{\theta(p-1)}})+\frac{|u|^{p-2}u|\nabla u|^{p}}{(1+|u|)^{\theta p}}=\mu,~&x\in\Omega,\\ u=0,~&x\in\partial\Omega,\end{array}\right.\end{equation*} 弱解的存在性和不存在性, 其中$\Omega\subseteq\mathbb{R}^N(N\geq3)$ 是有界光滑区域, $1相似文献   

一个局部波动方程组的解的全局存在性与爆破

我们研究初始值问题(e)u1/(e)t2=(e)2u1/(e)x2+‖u2(·,t)‖p, (e)2u2/(e)t2=(e)2u2/(e)x2+‖u1(·,t)‖q,-∞<x<∞,t>0,u1(x,0)=f1(x), (e)u1/(e)t(x,0)=g1(x),u2(x,0)=f2(x), (e)u2/(e)t(x,0)=g2(x),- ∞<x<∞,where‖ui(·,t)‖=∫∞-∞(4)i(x)|ui(x,t)|dx with (4)i(x)≥0 and ∫∞-∞(4)i(x)dx=1,i=1,2.然后建立解的全局存在和爆破的标准,提出爆破增长率.     

Semi-linear Wave Equations with Effective Damping

The authors study the Cauchy problem for the semi-linear damped wave equation $$u_{tt} - \Delta u + b\left( t \right)u_t = f\left( u \right), u\left( {0,x} \right) = u_0 \left( x \right), u_t \left( {0,x} \right) = u_1 \left( x \right)$$ in any space dimension n ≥ 1. It is assumed that the time-dependent damping term b(t) > 0 is effective, and in particular tb(t) → ∞ as t → ∞. The global existence of small energy data solutions for |f(u)| ≈ |u| ^p in the supercritical case of $p > \tfrac{2} {n}$ and $p \leqslant \tfrac{n} {{n - 2}}$ for n ≥ 3 is proved.     

一类带非局部源项的p-Laplace方程解的整体存在与爆破

本文研究一类在Neumann边值条件下带局部源项的p-Laplace方程解的整体存在和爆破性.利用微分不等式技巧,通过构造辅助函数的方法,获得了方程的解整体存在和解在有限时间爆破的充分条件,以及爆破时间的上下界估计,推广了相关文献结论.