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.     

一类具记忆项的非线性强阻尼双曲方程解的爆破性

利用一个特殊不等式得到一类具记忆项的非线性阻尼双曲方程解的爆破条件.方程描述了具记忆时滞的粘弹性梁问题.     

具粘性拟线性波方程的能量衰减估计

给出下列具粘性拟线性波方程初边值问题解的能量衰减估计u_(tt)(t,x)-div{σ(|▽u(t,x)|~2)▽u(t,x)}-△u(t,x)-△ut(t,x)+δ|u_t(t,x)|~(p-1)u_t(t,x)=μ|u(t,x)|~(q-1)u(t,x),x∈Ω,t∈(0,T),u(t,x)|■Ω=0,t∈(0,T),u(0,x)=u_0(x),u_t(0,x)=u_1(x),x∈Ω,其中Ω是R~N(N≥1)中具有光滑边界■Ω的区域,p≥1,q1,δ0,μ0,△表示Laplace算子,▽表示梯度算子和σ(s)是一给定的非线性函数.证明的思想是应用一已知的积分不等式,证明以上初边值问题解的能量衰减估计.     

一类退化非线性抛物型方程整体解的衰减估计

J.L.L ions用紧致性方法证明了一类退化非线性抛物型方程初边值问题整体解的存在唯一性,但解的衰减性很少有人考虑.应用M.N akao建立的差分不等式研究了整体解的衰减估计.     

一类四阶非线性波动方程解的爆破与衰减

本文研究了非线性阻尼项与源项的竞争对具有强阻尼项的四阶波动方程解的影响.利用不动点原理和势井方法给出了方程局部弱解存在唯一性满足的条件,证明了当mp且初始能量E(0)0时,解将在有限时间内爆破.同时对m,p的大小关系不加任何限制但存在t_0使0E(t_0)d的情况下,利用稳定集,研究了整体解的存在性,并得到了解的能量衰减估计.最后借助修正的能量泛函,指出当m≥p时弱解也是整体存在的,推广并改进了文献[1-6]中的结果.     

一类非线性波方程初边值问题解的爆破

该文研究如下具有非线性阻尼项和非线性源项的波方程的初边值问题 u_tt -u_xxt -u_xx -(σ(u²_x)u_x)_x+δ|u_t|^p-1u_t=μ|u|^q-1u, 0 < x <1, 0≤ t ≤T, (0.1) u(0, t)=u(1, t)=0, 0≤t≤ T, (0.2) u(x, 0)=u₀(x), u_t(x, 0)=u₁(x),0≤x≤1.(0.3) 文章将给出问题(0.1)--(0.3)的解在有限时刻爆破的充分条件, 同时将证明问题的局部广义解和局部古典解的存在性和唯一性.     

任意维数的强阻尼非线性波动方程（Ⅰ）—初边值问题

本文研究任意维数的强阻非线性波动方程ｕｔｔ－ａΔｕｔ－Δｕ＝ｆ（ｕ）具第一类齐边界条件的初值问题，设ｆ∈Ｃ＾１，ｆ＾１（ｕ）上方有界，且当ｎ≥４时存在常数Ａ，Ｂ和ｐ，使｜ｆ＾１（ｕ）｜≤Ａ｜ｕ｜＾ｐ＋Ｂ，其中０＜ｐ≤４／（ｎ－４）（ｎ＞４）：０＜ｐ＜∞（ｎ＝４），得到唯一整体强解，从而改进和推广了已知结果。     

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     

一类加权Kirchhoff方程非线性Liouville定理

本文侧重研究一类加权Kirchhoff方程弱解及稳定解的非线性Liouville型定理.利用适当构造试验函数技巧,当非线性函数满足适当条件时,我们在加权函数空间中证明了该方程弱解的非存在性.同时,当非线性函数为指数型且加权函数满足适当条件时,建立了方程稳定解的非存在性结论.     

Remarks on Blow-Up Phenomena in p-Laplacian Heat Equation with Inhomogeneous Nonlinearity

We investigate the $p$-Laplace heat equation $u_t-Delta_p u=ζ(t)f(u)$ in a bounded smooth domain. Using differential-inequality arguments, we prove blow-up results under suitable conditions on $ζ, f$, and the initial datum $u_0$. We also give an upper bound for the blow-up time in each case.     

Energy Decay of Solution to a Nonlinear Wave Equation of Kirchhoff Type

The paper studies the asymptotic behavior of solution to the initial boundary value problem for a nonlinear hyperbolic equation of Kirchhoff type It proves the global existence of solution by constructing a stable set and the energy exponential decayestimate by applying a lemma of V Komornik.     

Blow-up of the Solution for a Class of Porous Medium Equation with Positive Initial Energy

This paper deals with a class of porous medium equation

ut = Δum + f(u)$ut\hspace{0.17em}=\hspace{0.17em}\text{Δ}{u}^m\hspace{0.17em}+\hspace{0.17em}f\left(u\right)$

with homogeneous Dirichlet boundary conditions. The blow-up criteria is established by using the method of energy under the suitable condition on the function f(u).     

Exponential energy decay of solutions for a system of viscoelastic wave equations of Kirchhoff type with strong damping

The initial boundary value problem for a system of viscoelastic wave equations of Kirchhoff type with strong damping is considered. We prove that, under suitable assumptions on relaxation functions and certain initial data, the decay rate of the solutions energy is exponential.     

Nontrivial Solution for a Kirchhoff Type Problem with Zero Mass

Consider the Kirchhoff type equation \begin{equation}\label{eq0.1}-\left(a+b\int_{\mathbb{R}^{N}}|\nabla u|^{2}\,dx\right) \Delta u=\left(\frac{1}{|x|^\mu}*F(u)\right)f(u)\ \ \mbox{in}\ \mathbb{R}^N, \ \ u\in D^{1,2}(\mathbb{R}^N), ~~~~~~(0.1)\end{equation}where $a>0$, $b\geq0$, $0<\mu<\min\{N, 4\}$ with $N\geq 3$, $f: \mathbb{R}\to\mathbb{R}$ is a continuous function and $F(u)=\int_0^u f(t)\,dt$. Under some general assumptions on $f$, we establish the existence of a nontrivial spherically symmetric solution for problem (0.1). The proof is mainly based on mountain pass approach and a scaling technique introduced by Jeanjean.     

Blow-Up of Solutions for a Singular Nonlocal Viscoelastic Equation

We study the nonlinear one-dimensional viscoelastic nonlocal problem： uff-1/x（xux）x＋∫t0g（t-s）1/x）xux（x,s））xds=｜u｜p-2u,
with a nonlocal boundary condition. By the method given in [1, 2], we prove that there are solutions, under some conditions on the initial data, which blow up in finite time with nonpositive initial energy as well as positive initial energy. Estimates of the lifespan of blow-up solutions are also given. We improve a nonexistence result in Mesloub and Messaoudi [3].     

一类Kirchhoff型波动方程全局吸引子的存在性

利用Galerkin方法,研究了一类具有结构阻尼的kirchhoff型波动方程,方程是截面弹性杆运动的模型.通过各种不等式技巧及算子半群理论,证明了方程的解半群具有全局吸引子.     

Bounds for the Blow-Up Time on the Pseudo-Parabolic Equation with Nonlocal Term

We investigate the initial boundary value problem of the pseudo-parabolic equation $u_{t} - triangle u_{t} - triangle u = phi_{u}u + |u|^{p - 1}u,$ where $phi_{u}$ is the Newtonian potential, which was studied by Zhu et al. (Appl. Math. Comput., 329 (2018) 38-51), and the global existence and the finite time blow-up of the solutions were studied by the potential well method under the subcritical and critical initial energy levels. We in this note determine the upper and lower bounds for the blow-up time. While estimating the upper bound of blow-up time, we also find a sufficient condition of the solution blowing-up in finite time at arbitrary initial energy level. Moreover, we also refine the upper bounds for the blow-up time under the negative initial energy.     

R^N上一类Kirchhoff型方程径向对称正解的存在性

该文讨论了一类能量泛函不属于C^1类的Kirchhoff型方程,这类方程与等离子体物理和激光传输理论有密切的联系.通过变量变换,该文首先将所讨论的方程变成了与之等价的能量泛函属于C^1类的方程.然后,通过构造合适的Banach空间,在适当的条件下运用变分方法证明了所讨论的方程存在径向对称正解.