带有线性记忆的波方程在R~n上的时间依赖吸引子

本文基于Conti M,Di Plinio F等人提出的关于时间依赖全局吸引子的概念,研究了无界域上带有线性记忆的波方程解的长时间行为.利用尾部估计和压缩函数的方法证明了过程的渐近紧性,进而获得了H~1(R~n)×L~2(R~n)×L_u~2(R~+;H~1(R~n))上时间依赖吸引子的存在性.     

非线性弹性杆振动方程的全局吸引子及正则性

证明了非线性弹性杆振动方程全局吸引子的正则性,并进一步获得了(H_0~1(Ω)×H_0~1(Ω),(H~2(Ω)∩H_0~1(Ω))×(H~2(Ω)∩H_0~1(Ω)))-全局吸引子的存在性.     

三维Brinkman-Forchheimer方程强解的全局吸引子的存在性

研究了三维有界区域上Brinkman-Forchheimer方程■-γ△u+au+b|u|u+c|u|~βu+▽p=f强解的存在唯一性及强解的全局吸引子的存在性.首先证明了当5/2≤β≤4及初始值u_0∈H_0~1(Ω)时强解的存在唯一性.接着对强解进行了一系列一致估计,基于这些一致估计,借助半群理论证明了方程的强解分别在H_1~1(Ω)和H~2(Ω)空间中具有全局吸引子,并证明了H_0~1(Ω)中的全局吸引子实际上便是H~2(Ω)中的全局吸引子.     

边界层的奇性分析

设 λ∈[λ_0,∞)(0<λ_0<<1),H_1=H_0~2(Ω)∩H~3(Ω),H_2=H_0~1(Ω)∩H~3(Ω),H_3=H~3(Ω),k_1=1/4,k_2=1/12,k_3=1/36,J_6(λ)=integral d(x,Γ)≥a~λlog(1+a~(-β) |△▽(u_e-u)|~2dx,α(ε)=1/6×log_ε1/C(C>1).我们考虑问题(?)定理.若 u=f∈H_i,对问题(1),有如下三种情形成立:i)正规区域 当 λ_0≤λ≤1/6-α(ε)时,有J_6(λ)≤C‖f‖_(H~3(Ω))~2;ii)奇性增长区域当1/6-α(ε)<λ<1/6+k_i/6时,有J_6(λ)≤Cε~(-6λ+2k_i)‖f‖_(H~3(Ω))~2;iii)奇性稳定区域当 λ≥1/6+(k_i)/6时,有J_6(λ)≤Cε~(-1+k_i)‖f‖_(H~3(Ω))~2;其中 i=1,2,3,β≥(45)/(32),C 为同 ε 无关的常数(见图1).     

带有线性记忆的波方程在$\mathbb{R}^{n}$上的时间依赖吸引子

本文基于Conti M, Di Plinio F等人提出的关于时间依赖全局吸引子的概念, 研究了无界域上带有线性记忆的波方程解的长时间行为. 利用尾部估计和压缩函数的方法证明了过程的渐近紧性, 进而获得了$H^{1}(\mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})\times L^{2}_{\mu}(\mathbb{R}^{+};H^{1}(\mathbb{R}^{n}))$上时间依赖吸引子的存在性.     

具有时间依赖记忆核的非经典扩散方程的吸引子

该文在时间依赖空间H₀¹(Ω)×L_μt²(R⁺;H₀¹(Ω))中研究了具有时间依赖记忆核的非经典扩散方程解的长时间动力学行为.在新的理论框架下,利用积分估计方法以及分解技术得到了解的适定性,进而证明了时间依赖全局吸引子的存在性与正则性.     

带有Sobolev-Hardy临界指标项的非齐次椭圆方程的解

考虑带有Hardy和Sobolev-Hardy临界指标项的非齐次椭圆方程{-Δu-u(u/(|x|~2))=λu+(((|u|~(2~*(s)-2))/(|x|~s))u+f,在Ω中,u=0,在Ω上,这里2~*(s)=(2(N-s))/(N-2)是临界Sobolev-Hardy指标,N≥3,0≤s2,0≤μ=((N-2)~2)/4,ΩR~N是一个开区域.假设0≤λ≤λ_1时,λ_1是正算子-△-μ/(|x|~2)的第一特征值.f∈H~1_0(Ω)~*,f(x)≠0.当f满足适当的条件时,此方程在H~1_0(Ω)中至少具有两个解u_0和u_1.而且,当f≥0时,有u_0≥0和u_1≥0.     

反应扩散方程在H~2(Ω)和L~(2p-2)(Ω)中的指数吸引子

该文讨论一类具有任意多项式增长非线性项和非齐次项的反应扩散方程指数吸引子的存在性.首先,对R~3中的有界开子集Ω,分别选取解半群S(t)在L~2(Ω)和H~2(Ω)中的有界正不变吸收集来构造H~2(Ω)中的指数吸引子.然后证明对某个足够大的时间T_1,S(T_1)在这两个吸收集之间是Lipschitz连续的.最后由一种新的逼近技巧证明了对任意的g∈L~2(Ω),S(t)在L~(2p-2)(Ω)中存在指数吸引子.该结论推广了已有文献中的结果.     

带衰退记忆的经典反应扩散方程的全局吸引子

当非线性项满足任意阶多项式增长且外力项仅属于H~(-1)(Ω)时,研究了带衰退记忆的经典反应扩散方程的长时间动力学行为.应用抽象函数理论、半群理论以及新的估计技巧,在空间L~2(Ω)×L_μ~2(R~+;H_0~1(Ω))上证明了全局吸引子的存在性.该结果改进和推广了Chepyzhov等人(2006)及Zhong等人(2006)的相应结果.     

关于一类椭圆型方程的衔接问题

设区域Ω=Ω_1∪Ω_2∪Γ_0∪R~n,其中Ω_1,Ω_2为Ω的子区域,且,对一类一致椭圆型方程(或方程组)的边值问题,本文证明了,当原边值问题为适定时,新的衔接问题(由在Γ_0上满足衔接条件代替满足微分方程)是适定的,并且这二个问题的解是完全相同的。     

具有非线性记忆项的耗散波动方程的整体吸引子

LetΩRn be a bounded domain with a smooth boundary.We consider the longtime dynamics of a class of damped wave equations with a nonlinear memory term utt+αut-△u-∫0t 0μ(t-s)|u(s)| βu(s)ds + g(u)=f.Based on a time-uniform priori estimate method,the existence of the compact global attractor is proved for this model in the phase space H10(Ω)×L2(Ω).     

QUALITATIVE METHOD IN ANALYSIS OF A P-L-L WITH TANGENT CHARACTERISTIC AND FREQUENCY MODULATION INPUT

In analysis of p-L-L with tangent characteristic and frequency modulation input, we have obtained the following two types of the phase looked loop equation. \[\begin{array}{l} \frac{{{\partial ^2}\varphi }}{{\partial {t^2}}} + \alpha \frac{{d\varphi }}{{dt}} + \gamma \tan \varphi = {\beta _1} + {\beta _2}(\cos {\Omega _M}t + {\Omega _M}\sin {\Omega _M}t){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (I)\\frac{{{\partial ^2}\varphi }}{{\partial {t^2}}} + (\alpha + \eta {\sec ^2}\varphi )\frac{{d\varphi }}{{dt}} + \gamma \tan \varphi = {\beta _1} + {\beta _2}(\cos {\Omega _M}t - {\Omega _M}\sin {\Omega _M}t){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (II) \(\alpha > 0,\gamma > 0,\eta > 0,{\beta _1} > 0,{\beta _2} > 0,{\Omega _M} > 0) \end{array}\] In this paper, our aim is to explain the usual qualitative method and Lyapunov's function method, by which the existence of a periodic solution of (I), (II) is established. In addition, we especially point out: How is to construct the Lyapunovas function for the nonlinear and nonairtoiiomous system? This is a very important problem.     

Multiplicity of solutions for a class of quasilinear elliptic equation involving the critical Sobolev and Hardy exponents

In this paper, by using concentration-compactness principle and a new version of the symmetric mountain-pass lemma due to Kajikiya (J Funct Anal 225:352–370, 2005), infinitely many small solutions are obtained for a class of quasilinear elliptic equation with singular potential $$- \Delta_p u - \mu \frac{|u|^{p-2}u}{|x|^p} =\frac{|u|^{p^\ast(s)-2}u}{|x|^s} + \lambda f(x, u),\quad u\in H_0^{1,p}(\Omega).$$      

Extremal Functions for Adams Inequalities in Dimension Four

Let $\Omega\subset \mathbb{R}^4$ be a smooth bounded domain, $W_0^{2,2}(\Omega)$ be the usual Sobolev space. For any positive integer $\ell$, $\lambda_{\ell}(\Omega)$ is the $\ell$-th eigenvalue of the bi-Laplacian operator. Define $E_{\ell}=E_{\lambda_1(\Omega)}\oplus E_{\lambda_2(\Omega)}\oplus\cdots\oplus E_{\lambda_{\ell}(\Omega)}$, where $E_{\lambda_i(\Omega)}$ is eigenfunction space associated with $\lambda_i(\Omega)$. $E^{\bot}_{\ell}$ denotes the orthogonal complement of $E_\ell$ in $W_0^{2,2}(\Omega)$. For $0\leq\alpha<\lambda_{\ell+1}(\Omega)$, we define a norm by $\|u\|_{2,\alpha}^{2}=\|\Delta u\|^2_2-\alpha \|u\|^2_2$ for $u\in E^\bot_{\ell}$. In this paper, using the blow-up analysis, we prove the following Adams inequalities$$\sup_{u\in E_{\ell}^{\bot},\,\| u\|_{2,\alpha}\leq 1}\int_{\Omega}e^{32\pi^2u^2}{\rm d}x<+\infty;$$moreover, the above supremum can be attained by a function $u_0\in E_{\ell}^{\bot}\cap C^4(\overline{\Omega})$ with $\|u_0\|_{2,\alpha}=1$. This result extends that of Yang (J. Differential Equations, 2015), and complements that of Lu and Yang (Adv. Math. 2009) and Nguyen (arXiv: 1701.08249, 2017).     

Well-Posedness of Solutions for Sixth-Order Cahn-Hilliard Equation Arising in Oil-Water-Surfactant Mixtures

In this paper, by using the $L_p$-$L_q$-estimates, regularization property of the linear part of $e^{-t\Delta^3}$ and successive approximations, we consider the existence and uniqueness of global mild solutions to the sixth-order Cahn-Hilliard equation arising in oil-water-surfactant mixtures in suitable spaces, namely $C^0([0,T];\dot{W}^{2,\frac{N(l-1)}2}(\Omega))$ when the norm $\|u_0\|_{\dot{W}^{2,\frac{N(l-1)}2}(\Omega)}$ is sufficiently small.     

Multiplicity of Weak Solutions for a $(p(x), q(x))$-Kirchhoff Equation with Neumann Boundary Conditions

The aim of this study is to investigate the existence of infinitely many weak solutions for the $(p(x), q(x))$-Kirchhoff Neumann problem described by the following equation : \begin{equation*} \left\{\begin{array}{ll} -\left(a_{1}+a_{2}\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx\right)\Delta_{p(\cdot)}u-\left(b_{1}+b_{2}\int_{\Omega}\frac{1}{q(x)}|\nabla u|^{q(x)}dx\right)\Delta_{q(\cdot)}u\+\lambda(x)\Big(|u|^{p(x)-2} u+|u|^{q(x)-2} u\Big)= f_1(x,u)+f_2(x,u) &\mbox{ in } \Omega, \\frac{\partial u}{\partial \nu} =0 \quad &\mbox{on} \quad \partial\Omega.\end{array}\right. \end{equation*} By employing a critical point theorem proposed by B. Ricceri, which stems from a more comprehensive variational principle, we have successfully established the existence of infinitely many weak solutions for the aforementioned problem.     

局部分数阶拉普拉斯算子的狄利克雷问题弱解的细正则性

In this paper, we study the Holder regularity of weak solutions to the Dirichlet problem associated with the regional fractional Laplacian (-△)^αΩ on a bounded open set Ω ■R(N ≥ 2) with C^(1,1) boundary ■Ω. We prove that when f ∈ L^p(Ω), and g ∈ C(Ω), the following problem (-△)^αΩu = f in Ω, u = g on ■Ω, admits a unique weak solution u ∈ W^(α,2)(Ω) ∩ C(Ω),where p >N/2-2α and 1/2< α < 1. To solve this problem, we consider it into two special cases, i.e.,g ≡ 0 on ■Ω and f ≡ 0 in Ω. Finally, taking into account the preceding two cases, the general conclusion is drawn.     

Relaxation of variational problems in two-dimensional nonlinear elasticity

Consider a plate occupying in a reference configuration a bounded open set Ω ⊂ ℝ ² , and let be its stored-energy function. In this paper we are concerned with relaxation of variational problems of type:

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.     

粗糙核分数次积分算子的多线性算子在Hardy空间上的有界性

该文证明带有粗糙核的分数次积分算子的多线性算子\[T_{\Omega,\alpha}^{A}(f)(x)={\rm {\rm p.v.}}\int_{R^{n}}P_{m}(A;x,y)\frac{\Omega(x-y)}{|x-y|^{n-\alpha+m-1}}f(y){\rm d}y\]的$(H^{1}(\rr^{n}),L^{\frac{n}{n-\alpha},\infty}(\rr^{n}))$有界性.