Upper Semicontinuity Of Global Attractors Research Articles

Asymptotic analysis and upper semicontinuity with respect to delay term of attractors to binary mixtures of solids

We investigated the asymptotic dynamics of a nonlinear system modeling binary mixture of solids with delay term. Using the recent quasi-stability methods introduced by Chueshov and Lasiecka, we prove the existence, smoothness and finite dimensionality of a global attractor. We also prove the existence of exponential attractors. Moreover, we study the upper semicontinuity of global attractors with respect to small perturbations of the delay terms.

Long-time dynamical behavior for a piezoelectric system with magnetic effect and nonlinear dampings

<abstract><p>This paper is concerned with the long-time dynamical behavior of a piezoelectric system with magnetic effect, which has nonlinear damping terms and external forces with a parameter. At first, we use the nonlinear semigroup theory to prove the well-posedness of solutions. Then, we investigate the properties of global attractors and the existence of exponential attractors. Finally, the upper semicontinuity of global attractors has been investigated.</p></abstract>

Robustness of Global Attractors for Extensible Coupled Suspension Bridge Equations with Fractional Damping

In this paper we study the long-time dynamics of the perturbed system of suspension bridge equations $$\begin{aligned} m_c u_{tt}-\beta u_{xx}+\kappa (u-w)^+ +f_1(u,w)+ (-\partial _{xx})^{\gamma }u_{t}= & {} 0,\\ m_b w_{tt}+\mu w_{xxxx}+(p-\epsilon \Vert w_x\Vert ^2)w_{xx}-\kappa (u-w)^++f_2(u,w)+ (-\partial _{xx})^{\gamma }w_{t}= & {} 0, \end{aligned}$$ where $$\epsilon \in (0, 1]$$ is a perturbed parameter and $$\gamma \in (0, 1)$$ is said to be a fractional exponent. Under quite general assumptions on source terms and based on semigroup theory, we establish the global well-posedness and the existence of global attractors with finite fractal dimension. We analysis the upper semicontinuity of global attractors on the perturbed parameter $$\epsilon $$ in some sense. Moreover, we demonstrate an explicit control over semidistances between trajectories in the weak energy phase space in terms of $$\epsilon $$ . Finally, we prove that the family of global attractors is upper semicontinuous with respect to the fractional exponent $$\gamma \in (0,1/2)$$ .

Long-time dynamics for a fractional piezoelectric system with magnetic effects and Fourier's law

In this work, we use a variational approach to model vibrations on a piezoelectric beam with fractional damping depending on a parameter ν∈(0,1/2). Magnetic and thermal effects are taken into account via the Maxwell's equations and Fourier law, respectively. Existence and uniqueness of solutions of the system is proved by the semigroup theory. The existence of smooth global attractors with finite fractal dimension and the existence of exponential attractors for the associated dynamical system are proved. Finally, the upper-semicontinuity of global attractors as ν→0+ is shown.

Asymptotic behavior for a class of viscoelastic equations with memory lacking instantaneous damping

<abstract> In this paper, we mainly investigate long-time behavior for viscoelastic equation with fading memory <p class="disp_formula">$ u_{tt}-\Delta u_{tt}-\nu \Delta u+\int_{0}^{+\infty}k'(s)\Delta u(t-s)ds+f(u) = g(x). $ The main feature of the above equation is that the equation doesn't contain $ -\Delta u_t $, which contributes to a strong damping. The existence of global attractors is obtained by proving asymptotic compactness of the semigroup generated by the solutions for the viscoelastic equation. In addition, the upper semicontinuity of global attractors also is obtained. </abstract>

Quasi‐stability and upper semicontinuity for coupled parabolic equations with memory

AbstractThis current study deals with the long‐time dynamics of a nonlinear system of coupled parabolic equations with memory. The system describes the thermodiffusion phenomenon where the fluxes of mass diffusion and heat depend on the past history of the chemical potential and the temperature gradients, respectively, according to Gurtin‐Pipkin's law. Inspired by the works of Chueshov and Lasiecka on the property of quasi‐stability of dynamic systems, we prove this property for the problem considered in this study. This property allows us to analyze certain properties of global and exponential attractors in a more efficient and practical way. This approach is applied for the first time for coupled parabolic equations. We analyze the continuity of global attractors with respect to a pair of parameters in a residual dense set and their upper semicontinuity in a complete metric space. Finally, we analyze the upper semicontinuity of global attractors with respect to small perturbations of the damping terms.

Robustness of global attractors of singularly perturbed plate equations

In the paper, upper semicontinuity of global attractors of singularly perturbed plate equations on an unbounded domain with small positive parameter is considered. Under suitable assumptions, the equations possess a family of global attractors in natural energy space, and the corresponding singular limit equation, i.e. the parabolic equation, possesses a global attractor, which can be naturally embedded into a compact set of the natural energy space, and the upper semicontinuity of the family of global attractors to the compact set in the natural energy space (even more regular space) with respect to the Hausdorff semidistance, as the perturbation parameter tends to zero, was proved.

Upper Semicontinuity of Global Attractors for Singularly Perturbed Plate Equations

In the paper, upper semicontinuity of global attractors of singularly perturbed plate equations on an unbounded domain with small positive parameter, is considered. Under suitable assumptions, the equations possess a family of global attractors in natural energy space, and the corresponding singular limit equation, i.e., the parabolic equation, possesses a global attractor, which can be naturally embedded into a compact set of the natural energy space. Using the idea of tail estimates, the author established the upper semicontinuity of the family of global attractors to the compact set in the natural energy space (even more regular space) with respect to the Hausdorff semidistance, as the perturbation parameter tends to zero. The results obtained are new and are not common in the existing literature.

Upper semicontinuity of global attractors for a viscoelastic equations with nonlinear density and memory effects

This paper is devoted to showing the upper semicontinuity of global attractors associated with the family of nonlinear viscoelastic equations urn:x-wiley:mma:media:mma5389:mma5389-math-0001 in a three‐dimensional space, for f growing up to the critical exponent and dependent on ρ ∈ [0,4), as ρ→0+. This equation models extensional vibrations of thin rods with nonlinear material density ϱ(∂tu) = |∂tu|ρ and presence of memory effects. This type of problems has been extensively studied by several authors; the existence of a global attractor with optimal regularity for each ρ ∈ [0,4) were established only recently. The proof involves the optimal regularity of the attractors combined with Hausdorff's measure.

Upper semicontinuity of global attractors for a family of semilinear wave equations with gentle dissipation

The paper investigates the upper semicontinuity of global attractors Aα of a family of semilinear wave equations with gentle dissipation: utt−△u+γ(−△)αut+f(u)=g(x), with α∈(0,1∕2). It is a continuation of researches in Yang et al. (2016), where the existence of global attractors of the equations has been established. We further show that for any neighborhood U of Aα0,α0∈(0,1∕2),Aα⊂U when 0<α−α0≪1.

On the upper semicontinuity of global attractors for damped wave equations

We provide a new proof of the upper-semicontinuity property for the global attractorsadmitted by the solution operators associated with some strongly damped wave equations. In particular,we demonstrate an explicit control over semidistances between trajectories in the weak energy phasespace in terms of the perturbation parameter. This result strengthens the recent work by Y. Wang andC. Zhong [7].

Upper semicontinuity of global attractors for quasilinear parabolic equations on unbounded thin domains

We consider the asymptotic behavior of quasilinear parabolic equations posed in a family of unbounded domains that degenerates onto a lower dimensional set. Considering an auxiliary family of weighted Sobolev spaces we show the existence of global attractors and we analyze convergence properties of the solutions as well of the attractors.

Upper semicontinuity of global attractors for damped wave equations

Upper semicontinuity of global attractors for damped wave equations

Upper semicontinuity of attractors for nonclassical diffusion equations in [formula omitted

In this paper, we mainly investigate upper semicontinuity of global attractors in H1(R3) for a perturbed nonclassical diffusion equations. For this purpose, some necessary abstract results are established.

Existence and upper semicontinuity of global attractors for a $p$-Laplacian inclusion

In this work we study the asymptotic behavior of a $p$-Laplacianinclusion of the form $\displaystyle\frac{\partialu_\lambda}{\partial t} - div(D^\lambda|\nablau_\lambda|^{p-2}\nabla u_\lambda) + |u_\lambda|^{p-2}u_\lambda$ $\in F(u_\lambda) + h,$ where $p&gt;2$, $h\in L^2(\Omega),$ with$\Omega\subset\mathbb{R}^n,\; n\geq 1,$ a bounded smooth domain,$D^\lambda \in L^\infty(\Omega)$, $\infty &gt; M\geq D^\lambda(x)\geq \sigma &gt;0$ a.e. in $\Omega$, $\lambda \in [0,\infty)$ and$D^\lambda\rightarrow D^{\lambda_1}$ in $L^\infty(\Omega)$ as$\lambda \to \lambda_1$, $F:\mathcal{D}(F)\subsetL^{2}(\Omega)\rightarrow\mathcal{P}(L^{2}(\Omega))$, given by$F(y(\cdot))=\{\xi(\cdot)\in L^{2}(\Omega):\xi(x)\inf(y(x))\;x\mbox{-a.e. in}\; \Omega\}$ with$f:\mathbb{R}\rightarrow\mathcal{C}_{v}(\mathbb{R})$ Lipschitz($\mathcal{C}_{v}(\mathbb{R})$ is the set of all nonempty,bounded, closed, convex subsets of $\mathbb{R}$) be a multivaluedmap. We prove the existence of a global attractor in $L^2(\Omega)$for each positive finite diffusion coefficient and we show thatthe family of attractors behaves upper semicontinuously onpositive finite diffusion parameters.

Continuity of the flows and upper semicontinuity of global attractors for [formula omitted]-Laplacian parabolic problems

In this work we prove continuity of solutions with respect to initial conditions and exponent parameters and we prove upper semicontinuity of a family of global attractors for one-dimensional problems of the form ∂us∂t−∂∂x(|∂us∂x|ps(x)−2∂us∂x)=B(us) where B is a globally Lipschitz map, ps(⋅)→pinL∞(I)(I:=(c,d)andp>2constant) as s goes to infinity.

Existence and upper semicontinuity of global attractors for [formula omitted]-Laplacian systems

This work concerns the study of asymptotic behavior of coupled systems of p ( x ) -Laplacian differential inclusions. We obtain that the generalized semiflow generated by the coupled system has a global attractor for each pair of positive diffusion coefficients. We also prove that the attractors are upper semicontinuous on positive finite diffusion parameters. In particular, to get the results we prove that the p ( x ) -Laplacian operator is subdifferential of a convex, proper and lower semicontinuous non-negative map.

Existence and Upper Semicontinuity of Global Attractors for Neural Network in a Bounded Domain

In this work we prove the existence of a compact global attractor for the flow of the equation $$\frac{\partial m(r,t)}{\partial t}=-m(r,t)+ J*(f\circ m)(r,t)+ h, \,\,\, h, > 0,$$ in L 2(S 1). We also give uniform estimates on the size of the attractor and show that the family of attractors $${\{{\mathcal A}_{J}\}}$$ is upper semicontinuous at J 0.

Regularity of trajectory attractor and upper semicontinuity of global attractor for a 2D non-Newtonian fluid

There are two results within this paper. The one is the regularity of trajectory attractor and the trajectory asymptotic smoothing effect of the incompressible non-Newtonian fluid on 2D bounded domains, for which the solution to each initial value could be non-unique. The other is the upper semicontinuity of global attractors of the addressed fluid when the spatial domains vary from Ω m to Ω = R × ( − L , L ) , where { Ω m } m = 1 ∞ is an expanding sequence of simply connected, bounded and smooth subdomains of Ω such that Ω m → Ω as m → + ∞ . That is, let A and A m be the global attractors of the fluid corresponding to Ω and Ω m , respectively, we establish that for any neighborhood O ( A ) of A , the global attractor A m enters O ( A ) if m is large enough.

Upper semicontinuity of global attractors for the perturbed viscous Cahn-Hilliard equations

It is known that the semigroup generated by the initial-boundary value problem for the perturbed viscous Cahn-Hilliard equation with $\varepsilon> 0$ as a parameter admits a global attractor $\mathcal{A}_{\varepsilon}$ in the phase space $X^{{1}/{2}} =(H^2(\Omega)\cap H^{1}_{0}(\Omega))\times L^2(\Omega)$, $\Omega\subset \mathbb{R}^n$, $n\leq 3$ (see [M. B. Kania, Global attractor for the perturbed viscous Cahn-Hilliard equation , Colloq. Math. 109 (2007), 217-229]). In this paper we show that the family $\{\mathcal{A}_{\varepsilon}\}_{\varepsilon\in[0,1]}$ is upper semicontinuous at $0$, which means that the Hausdorff semidistance $$ d_{X^{{1}/{2}}}(\mathcal{A}_{\varepsilon},\mathcal{A}_0)\equiv \sup_{\psi\in \mathcal{A}_{\varepsilon}}\inf_{\phi\in\mathcal{A}_{0}}\| \psi-\phi\|_{X^{{1}/{2}}}, $$ tends to 0 as $\varepsilon\to 0^{+}$.