一类非自治Brinkman-Forchheimer系统的拉回$\mathcal{D}$-吸引子

The asymptotic behavior of solutions of the three-dimensional nonautonomous Brinkman-Forchheimer equation is investigated. And the existence of pullback global attractors in L2 (Ω) and H01(Ω) is proved, respectively.     

Random attractors for non-autonomous fractional stochastic Ginzburg-Landau equations on unbounded domains

This paper deals with the dynamical behavior of solutions for non-autonomous stochastic fractional Ginzburg-Landau equations driven by additive noise with $\alpha\in(0,1)$. We prove the existence and uniqueness of tempered pullback random attractors for the equations in $L^{2}(\mathbf{R}^{3})$. In addition, we also obtain the upper semicontinuity of random attractors when the intensity of noise approaches zero. The main difficulty here is the noncompactness of Sobolev embeddings on unbounded domains. To solve this, we establish the pullback asymptotic compactness of solutions in $L^{2}(\mathbf{R}^{3})$ by the tail-estimates of solutions.     

非自治Plate方程时间依赖强拉回吸引子的存在性

结合Plinio等人([Plinio D,Duane G S,Temarn R,Time-dependent attractor for the oscillon equation,Discrete Contin Dyn Syst,2011,29(1):141-167.])提出的时间依赖全局吸引子概念,运用压缩函数的方法,证明了带有时间依赖系数的非自治Plate方程时间依赖拉回吸引子在空间H~4(Ω)∩H~2_0(Ω)×H~2_0(Ω)中的存在性.     

The Upper Semicontinuity of Random Attractors for Non-Autonomous Stochastic Plate Equations with Multiplicative Noise and Nonlinear Damping

Based on the existence of pullback attractors for the non-autonomous stochastic plate equations with multiplicative noise and nonlinear damping defined in the entire space $\mathbb{R}^n$ by Xiaobin Yao in \cite{Yao4}, in the paper, we further investigate the upper semicontinuity of pullback attractors for the problem.     

Regularity of pullback attractors for non-autonomous stochastic coupled reaction-diffusion systems

We consider the dynamical behavior of the typical non-autonomous autocatalytic stochastic coupled reaction-diffusion systems on the entire space $\mathbb{R}^n$. Some new uniform asymptotic estimates are implemented to investigate the existence of pullback attractors in the Sobolev space $H^1(\mathbb{R}^n)^3$ for the three-component reversible Gray-Scott system.     

Pullback exponential attractors for nonautonomous dynamical system in space of higher regularity

Under what condition, a process which exists a $(E,E)$-pullback exponential attractor implies the existence of $(E,V)$- pullback exponential attractor when $V$ embedded in $E$? We answer this question in this paper. As an application of this result, we prove the existence of pullback exponential attractor for a nonlinear reaction-diffusion equation with a polynomial growth nonlinearity in $L^q(\Omega)(\forall q\geq 2)$ and $H_0^1(\Omega)$.     

Regularity of pullback attractors and random equilibrium for non-autonomous stochastic FitzHugh-Nagumo system on unbounded domains

This paper is concerned with the stochastic Fitzhugh-Nagumo system with non-autonomous terms as well as Wiener type multiplicative noises. By using the so-called notions of uniform absorption and uniformly pullback asymptotic compactness, the existences and upper semi-continuity of pullback attractors are proved for the generated random cocycle in $L^l(\mathbb{R}^N)\times L^2(\mathbb{R}^N)$ for any $l\in(2,p]$. The asymptotic compactness of the first component of the system in $L^p(\mathbb{R}^N)$ is proved by a new asymptotic a priori estimate technique, by which the plus or minus sign of the nonlinearity at large values is not required. Moreover, the condition on the existence of the unique random fixed point is obtained, in which case the influence of physical parameters on the attractors is analysed.     

The second order pullback equation

Let $f,g$ be two closed $k$ -forms over $\mathbb{R }^{n}.$ The pullback equation studies the existence of a diffeomorphism $\varphi :\mathbb{R }^{n} \rightarrow \mathbb{R }^{n}$ such that $$\begin{aligned} \varphi ^{*}(g)=f. \end{aligned}$$ We prove two types of results. The first one sharpens some of the existing regularity results. The second one discusses the possibility of choosing the map $\varphi $ as the gradient of a function $\Phi :\mathbb{R }^{n} \rightarrow \mathbb R .$ We show that this is a very rare event unless the two forms are constant.     

Inequalities and Separation for a Biharmonic Laplace-Beltrami Differential Operator in a Hilbert Space Associated with the Existence and Uniqueness Theorem

In this paper, we have studied the separation for the biharmonic Laplace-Beltrami differential operator\begin{equation*}Au(x)=-\Delta \Delta u(x)+V(x)u(x),\end{equation*}for all $x\in R^{n}$, in the Hilbert space $H=L_{2}(R^{n},H_{1})$ with the operator potential $V(x)\in C^{1}(R^{n},L(H_{1}))$, where $L(H_{1})$ is the space of all bounded linear operators on the Hilbert space $H_{1}$, while $\Delta \Delta u$\ is the biharmonic differential operator and\begin{equation*}\Delta u{=-}\sum_{i,j=1}^{n}\frac{1}{\sqrt{\det g}}\frac{\partial }{{\partial x_{i}}}\left[ \sqrt{\det g}g^{-1}(x)\frac{\partial u}{{\partial x}_{j}}\right]\end{equation*}is the Laplace-Beltrami differential operator in $R^{n}$. Here $g(x)=(g_{ij}(x))$ is the Riemannian matrix, while $g^{-1}(x)$ is the inverse of the matrix $g(x)$. Moreover, we have studied the existence and uniqueness Theorem for the solution of the non-homogeneous biharmonic Laplace-Beltrami differential equation $Au=-\Delta \Delta u+V(x)u(x)=f(x)$ in the Hilbert space $H$ where $f(x)\in H$ as an application of the separation approach.     

The existence of pullback exponential attractors for nonautonomous dynamical system and applications to nonautonomous reaction diffusion equations

First we establish some sufficient conditions for the existence of pullback exponential attractors by using $\omega-$limit compactness in the framework of process. Then we provide a new method to prove the existence of pullback exponential attractors. As a simple application, we prove the existence of pullback exponential attractors for nonautonomous reaction diffusion equations in $H_0^1$.     

带参数的一阶泛函差分方程的定号周期解

In this paper,the authors obtain the existence of one-signed periodic solutions of the first-order functional difference equation ?u(n) = a(n)u(n)-λb(n)f(u(n-τ(n))),n ∈ Z by using global bifurcation techniques,where a,b:Z → [0,∞) are T-periodic functions with ∑T n=1 a(n) 0,∑T n=1 b(n) 0;τ:Z → Z is T-periodic function,λ 0 is a parameter;f ∈ C(R,R) and there exist two constants s_2 0 s_1 such that f(s_2) = f(0) = f(s_1) = 0,f(s) 0 for s ∈(0,s_1) ∪(s_1,∞),and f(s) 0 for s ∈(-∞,s_2) ∪(s_2,0).     

非自治FitzHugh-Nagumo系统拉回吸引子的H2\times H1_0 有界性

研究带奇异扰动非自治~FitzHugh-Nagumo系统拉回吸引子的 $H^{2}\times H^{1}_{0}$ 有界性. 为此, 首先建立关于过程有界不变集的 $H^{2}\times H^{1}_{0}$ 有界性, 从而得到原系统拉回吸引子的有界性结果.     

On solvability of a class of nonlinear elliptic type equation with variable exponent

In this paper, we study the Dirichlet problem for the implicit degenerate nonlinear elliptic equation with variable exponent in a bounded domain $\Omega \subset \mathbb{R}^{n}$. We obtain sufficient conditions for the existence of a solution without regularization and any restriction between the exponents. Furthermore, we define the domain of the operator generated by posed problem and investigate its some properties and also its relations with known spaces that enable us to prove existence theorem.     

Dynamics of Stochastic Ginzburg-Landau Equations Driven by Colored Noise on Thin Domains

This work is concerned with the asymptotic behaviors of solutions to a class of non-autonomous stochastic Ginzburg-Landau equations driven by colored noise and deterministic non-autonomous terms defined on thin domains. The existence and uniqueness of tempered pullback random attractors are proved for the stochastic Ginzburg-Landau systems defined on $(n+1)$-dimensional narrow domain. Furthermore, the upper semicontinuity of these attractors is established, when a family of $(n+1)$-dimensional thin domains collapses onto an $n$-dimensional domain.     

Wong-Zakai approximations and attractors for fractional stochastic reaction-diffusion equations on unbounded domains

In this paper, we investigate the Wong-Zakai approximations induced by a stationary process and the long term behavior of the fractional stochastic reaction-diffusion equation driven by a white noise. Precisely, one of the main ingredients in this paper is to establish the existence and uniqueness of tempered pullback attractors for the Wong-Zakai approximations of fractional stochastic reaction-diffusion equations. Thereafter the upper semi-continuity of attractors for the Wong-Zakai approximation of the equation as $\delta\rightarrow0$ is proved.     

三维拟线性波方程的小初值光滑解

对三维小初值拟线性波方程3∑(i,j=0)g~(ij)(u)■_(ij)u=0,H.Lindblad证明了它有整体光滑解.本文考虑如下带有小初值的拟线性波方程3∑(i,j=0)g~(ij)(u)■_(ij)u=(■u)~3,通过得到低阶导数的衰减估计和高阶导数的能量估计,由连续论证法证明了这个方程也存在整体光滑解.     

对流扩散方程的解

关注如下的对流扩散方程 $$ u_{t}=\text{div}(|\nabla u^{m}|^{p-2}\nabla u^{m})+\sum_{i=1}^{N}\frac{\partial b_{i}(u^{m})}{\partial x_{i}} $$ 的初边值问题. 若 $p>1+\frac{1}{m}$, 通过考虑正则化问题的解 $u_{k}$, 利用 Moser 迭代技巧, 得到了$u_{k}$ 的 $L^{\infty}$ 模与 梯度 $\nabla u_{k}$ 的 $L^{p}$ 模的局部有界性. 利用紧致性定理, 得到了对流扩散方程本身解的存在性. 若 $p<1+\frac{1}{m},\ p>2$ 或者 $p=1+\frac{1}{m}$, 利用类似的方法可以得到解的存在性. 证明了解的唯一性, 同时讨论了正性和熄灭性等解的性质.     

New Class of Kirchhoff Type Equations with Kelvin-Voigt Damping and General Nonlinearity: Local Existence and Blow-up in Solutions

In this paper, we consider a class of Kirchhoff equation, in the presence of a Kelvin-Voigt type damping and a source term of general nonlinearity forms. Where the studied equation is given as follows\begin{equation*}u_{tt} -\mathcal{K}\left( \mathcal{N}u(t)\right)\left[ \Delta_{p(x)}u +\Delta_{r(x)}u_{t}\right]=\mathcal{F}(x, t, u).\end{equation*}Here, $\mathcal{K}\left( \mathcal{N}u(t)\right)$ is a Kirchhoff function, $\Delta_{r(x)}u_{t}$ represent a Kelvin-Voigt strong damping term, and $\mathcal{F}(x, t, u)$ is a source term. According to an appropriate assumption, we obtain the local existence of the weak solutions by applying the Galerkin's approximation method. Furthermore, we prove a non-global existence result for certain solutions with negative/positive initial energy. More precisely, our aim is to find a sufficient conditions for $p(x), q(x), r(x), \mathcal{F}(x,t,u)$ and the initial data for which the blow-up occurs.     

由H\"{o}rmander向量场构成的抛物方程的 $W_{\ast}^{1,p}$ 正则性

设 $X_{1},\cdots ,X_{q}\ (q相似文献   

具有结构阻尼和外阻尼的非自治非线性梁方程的拉回D_δ,E_1-吸引子

考虑了具有结构阻尼和外阻尼的非自治非线性粘弹性梁方程的拉回D_δ,E_1-吸引子.首先利用Galerkin方法,证明了在齐次边界条件和初始条件下系统在V×H和D(A)×V中的整体解的存在唯一性;其次通过先验估计,证明了系统的拉回吸收集的存在性;最后证明了系统满足拉回D_δ,E_1-条件(C),从而证明了系统的强拉回D_δ,E_1-吸引子的存在性.