Regularity Of Global Attractor Research Articles

Upper semicontinuity of optimal attractors for viscoelastic equations lacking strong damping

ABSTRACT This paper devotes to investigating the asymptotic behavior of viscoelastic equation with fading memory lacking strong damping term where . Its key feature is the lack of strong mechanical damping term replaced by weaker dissipative term (i.e. memory term). At present, the existing results only considered the existence and regularity of global attractors, but do not obtain the existence of the strong attractors, because the compactness is difficult to get in highly regular space. Accordingly, in current work, we will prove the existence of strongly global attractors with only weak dissipation when the nonlinearity f satisfies critical growth by using the dominated convergence theorem and the asymptotically contractive function method. Moreover, the upper semicontinuity of the strongly global attractors is also obtained when the perturbed parameter , which deepens some conclusions that exist in Conti et al. [Asymptotics of viscoelastic materials with nonlinear density and memory effects. J Differ Equ. 2018;264:4235–4259] and Qin et al. [Uniform attractors for a nonautonomous viscoelastic equation with a past history. Nonlinear Theory Methods Appl. 2014;101:1–15].

Regularity of global attractors and exponential attractors for $ 2 $D quasi-geostrophic equations with fractional dissipation

<p style='text-indent:20px;'>In this paper we investigate the regularity of global attractors and of exponential attractors for two dimensional quasi-geostrophic equations with fractional dissipation in <inline-formula><tex-math id="M2">\begin{document}$ H^{2\alpha+s}(\mathbb{T}^2) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M3">\begin{document}$ \alpha>\frac{1}{2} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ s>1. $\end{document}</tex-math></inline-formula> We prove the existence of <inline-formula><tex-math id="M5">\begin{document}$ (H^{2\alpha^-+s}(\mathbb{T}^2),H^{2\alpha+s}(\mathbb{T}^2)) $\end{document}</tex-math></inline-formula>-global attractor <inline-formula><tex-math id="M6">\begin{document}$ \mathcal{A}, $\end{document}</tex-math></inline-formula> that is, <inline-formula><tex-math id="M7">\begin{document}$ \mathcal{A} $\end{document}</tex-math></inline-formula> is compact in <inline-formula><tex-math id="M8">\begin{document}$ H^{2\alpha+s}(\mathbb{T}^2) $\end{document}</tex-math></inline-formula> and attracts all bounded subsets of <inline-formula><tex-math id="M9">\begin{document}$ H^{2\alpha^-+s}(\mathbb{T}^2) $\end{document}</tex-math></inline-formula> with respect to the norm of <inline-formula><tex-math id="M10">\begin{document}$ H^{2\alpha+s}(\mathbb{T}^2). $\end{document}</tex-math></inline-formula> The asymptotic compactness of solutions in <inline-formula><tex-math id="M11">\begin{document}$ H^{2\alpha+s}(\mathbb{T}^2) $\end{document}</tex-math></inline-formula> is established by using commutator estimates for nonlinear terms, the spectral decomposition of solutions and new estimates of higher order derivatives. Furthermore, we show the existence of the exponential attractor in <inline-formula><tex-math id="M12">\begin{document}$ H^{2\alpha+s}(\mathbb{T}^2), $\end{document}</tex-math></inline-formula> whose compactness, boundedness of the fractional dimension and exponential attractiveness for the bounded subset of <inline-formula><tex-math id="M13">\begin{document}$ H^{2\alpha^-+s}(\mathbb{T}^2) $\end{document}</tex-math></inline-formula> are all in the topology of <inline-formula><tex-math id="M14">\begin{document}$ H^{2\alpha+s}(\mathbb{T}^2). $\end{document}</tex-math></inline-formula></p>

Asymptotic behaviour of the solutions for a weakly damped anisotropic sixth-order Schrödinger type equation in $ \mathbb{R}^2 $

<p style='text-indent:20px;'>We study the long-time behaviour of the solutions to a nonlinear damped anisotropic sixth-order Schrödinger type equation in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^2 $\end{document}</tex-math></inline-formula> that reads</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_t+i\Delta u-i \left(\partial_y^4 u-\partial_y^6 u\right)+ig(|u|^2)u+\gamma u = f\,,\;\;(t,(x,y))\in \mathbb{R}\times \mathbb{R}^2\,. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>We prove that this behaviour is described by the existence of regular global attractor in an anisotropic Sobolev space with finite fractal dimension.</p>

Upper semicontinuity of attractors for nonclassical diffusion equations with arbitrary polynomial growth

In this paper, we mainly investigate upper semicontinuity and regularity of attractors for nonclassical diffusion equations with perturbed parameters ν and the nonlinear term f satisfying the polynomial growth of arbitrary order p-1 (p geq 2). We extend the asymptotic a priori estimate method (see (Wang et al. in Appl. Math. Comput. 240:51–61, 2014)) to verify asymptotic compactness and upper semicontinuity of a family of semigroups for autonomous dynamical systems (see Theorems 2.2 and 2.3). By using the new operator decomposition method, we construct asymptotic contractive function and obtain the upper semicontinuity for our problem, which generalizes the results obtained in (Wang et al. in Appl. Math. Comput. 240:51–61, 2014). In particular, the regularity of global attractors is obtained, which extends and improves some results in (Xie et al. in J. Funct. Spaces 2016:5340489, 2016; Xie et al. in Nonlinear Anal. 31:23–37, 2016).

Regular Global Attractors for Wave Equations with Degenerate Memory

We consider the wave equation with degenerate viscoelastic dissipation recently examined in Cavalcanti, Fatori, and Ma, Attractors for wave equations with degenerate memory , J. Differential Equations (2016). Under certain extra assumptions (namely on the nonlinear term), we show the existence of a compact attracting set which provides further regularity for the global attractor and show that it consists of regular solutions.

Global attractors for the 2D hyperbolic Cahn–Hilliard equations

We consider the initial boundary value problem for the hyperbolic relaxation of the 2D Cahn–Hilliard equation with sub-cubic nonlinearity. Under mild regularity conditions on the nonlinearity, we prove the uniform (with respect to the initial data) boundedness of the weak solutions without assuming lower bound condition on the first derivative of the nonlinear term. Then, we prove the existence of the regular global attractor for the weak solutions.

GLOBAL ATTRACTORS AND REGULARITY FOR THE EXTENSIBLE SUSPENSION BRIDGE EQUATIONS WITH PAST HISTORY

In this paper, we study the long-time dynamical behavior for the extensible suspension bridge equations with past history. We prove the existence of the global attractors by using the contraction function method. Furthermore, the regularity of global attractor is achieved.

Global attractors for the coupled suspension bridge system with temperature

This paper deals with the long‐term properties of the thermoelastic nonlinear string‐beam system related to the well‐known Lazer–McKenna suspension bridge model urn:x-wiley:mma:media:mma3526:mma3526-math-0001 In particular, no mechanical dissipation occurs in the equations, because the loss of energy is entirely due to thermal effects. The existence of regular global attractors for the associated solution semigroup is proved (without resorting to a bootstrap argument) for time‐independent supplies f,g,h and any . Copyright © 2015 John Wiley & Sons, Ltd.

On the Existence of Regular Global Attractor for $$p$$ p -Laplacian Evolution Equation

In this study, we consider the nonlinear evolution equation of parabolic type $$\begin{aligned} u_t-\mathrm{div}\big (|\nabla u|^{p-2}\nabla u\big )+f(u)=g. \end{aligned}$$ut-div(| u|p-2 u)+f(u)=g....

Higher regularity of global attractor for a damped Benjamin–Bona–Mahony equation on R

For one-dimensional damped Benjamin–Bona–Mahony equation, a recent result claimed that there exists a global attractor in , in fact the attractor is smoother and it belongs to for every . In this paper, by some techniques of harmonic analysis, we first show that the solution operator of one-dimensional damped BBM equation is asymptotically compact, then we give the existence of global attractor in . Finally, we obtain the higher regularity of the global attractor, it is in fact bounded in .

Well-posedness and longtime behaviour of a coupled nonlinear system modeling a suspension bridge

In this paper we discuss the well-posedness and the asymptotic behavior of a doubly nonlinear problem describing the vibrations of a Kelvin–Voigt string–beam system which models a suspension bridge. For this model we obtain the existence and uniqueness of solutions and the exponential stability of the homogeneous system, provided that the constant axial force \(p\) is smaller than a critical value. For a general \(p,\) the existence of the regular global attractor is proved when the external loads are independent of time.

Long-term dynamics of a viscoelastic suspension bridge

In this paper we scrutinize the asymptotic behavior of a nonlinear problem which models the vertical vibrations of a suspension bridge. The single-span road-bed is modeled as an extensible viscoelastic beam which is simply supported at the ends. It is suspended to a rigid and immovable frame by means of a distributed system of vertical one-sided elastic springs. A constant axial force p is applied at one end of the deck, and time-independent vertical loads are allowed. For this model we obtain original results, including the existence of a regular global attractor for all $${p\in\mathbb{R}.}$$ In spite of the extremely weak dissipation due to the convolution term, this result is achieved by exploiting the exponential decay of the memory kernel.

Regular attractors of autonomous and nonautonomous dynamical systems

This paper studies regular global attractors of the dynamical systems corresponding to dissipative evolu� tion equations and their nonautonomous perturba� tions. We prove that the nonautonomous dynamical system (process) resulting from a small nonautono� mous perturbation of an autonomous dynamical sys� tem (semigroup) having a regular attractor has a regu� lar nonautonomous attractor as well. Moreover, the symmetric Hausdorff deviation of the perturbed attractors from the unperturbed ones is bounded above by O(e κ ), where e is a perturbation parameter and 0 < κ < 1. We apply the obtained results to weakly dissipa� tive wave equations in a bounded domain in 3 per�

On the viscoelastic coupled suspension bridge

In this paper we discuss the asymptotic behavior of a doubly nonlinear problem describing the vibrations of a coupled suspension bridge. The single-span road-bed is modeled as an extensible viscoelastic beam which is simply supported at the ends. The main cable is modeled by a viscoelastic string and is connected to the road-bed by a distributed system of one-sided elastic springs. A constant axial force $p$ is applied at one end of the deck, and time-independent vertical loads are allowed to act both on the road-bed and on the suspension cable. For this general model we obtain original results, including the existence of a regular global attractor for all $p\in\mathbb{R}$.

Regular attractors and nonautonomous perturbations of them

We study regular global attractors of dissipative dynamical semigroups with discrete or continuous time and we investigate attractors for nonautonomous perturbations of such semigroups. The main theorem states that the regularity of global attractors is preserved under small nonautonomous perturbations. Moreover, nonautonomous regular global attractors remain exponential and robust. We apply these general results to model nonautonomous reaction-diffusion systems in a bounded domain of with time-dependent external forces.Bibliography: 22 titles.

On the Theory of Global Attractors and Lyapunov Functionals

From the point of view of longterm dynamics, we study multivalued and single-valued semigroups of operators acting on complete metric spaces. We provide necessary and sufficient conditions for the existence of the global attractor under minimal requirements in terms of continuity of the semigroup. In the case of single-valued semigroups possessing a Lyapunov functional, we exhibit a simple proof of the existence and the characterization of the attractor in terms of the unstable set of stationary points. As an application, we consider the multivalued semigroup generated by the equation ruling the evolution of the specific humidity in a system of moist air, and we prove the existence of a regular global attractor.

Strongly damped wave equations with critical nonlinearities

This note deals with the strongly damped nonlinear wave equation utt−Δut−Δu+f(ut)+g(u)=h with Dirichlet boundary conditions, where both the nonlinearities f and g exhibit a critical growth, while h is a time-independent forcing term. The existence of an exponential attractor of optimal regularity is proven. As a corollary, a regular global attractor of finite fractal dimension is obtained.

Regularity of Global Attractor for the Reaction‐Diffusion Equation

By using an iteration procedure, regularity estimates for the linear semigroups, and a classical existence theorem of global attractor, we prove that the reaction‐diffusion equation possesses a global attractor in Sobolev space Hk for all k &gt; 0, which attracts any bounded subset of Hk(Ω) in the Hk‐norm.

Two-dimensional reaction-diffusion equations with memory

In a two-dimensional space domain, we consider a reaction-diffusion equation whose diffusion term is a time convolution of the Laplace operator against a nonin-creasing summable memory kernel κ. This equation models several phenomena arising from many different areas. After rescaling κ by a relaxation time e > 0, we formulate a Cauchy-Dirichlet problem, which is rigorously translated into a similar problem for a semilinear hyperbolic integro-differential equation with nonlinear damping, for a particular choice of the initial data. Using the past history approach, we show that the hyperbolic equation generates a dynamical system which is dissipative provided that e is small enough, namely, when the equation is sufficiently close to the standard reaction-diffusion equation formally obtained by replacing κ with the Dirac mass at 0. Then, we provide an estimate of the difference between e-trajectories and 0-trajectories, and we construct a family of regular exponential attractors which is robust with respect to the singular limit e → 0. In particular, this yields the existence of a regular global attractor of finite fractal dimension. Convergence to equilibria is also examined. Finally, all the results are reinterpreted within the original framework.

Time-dependent attractor for the Oscillon equation

We investigate the asymptotic behavior of the nonautonomous evolution problem generated by the Oscillon equation ∂ tt $u(x,t) +H $ ∂ t$ u(x,t) -\e^{-2Ht}$ ∂ xx $ u(x,t) + V'(u(x,t)) =0, \quad (x,t)\in (0,1) \times \R,$ with periodic boundary conditions, where $H>0$ is the Hubble constant and $V$ is a nonlinear potential of arbitrary polynomial growth. After constructing a suitable dynamical framework to deal with the explicit time dependence of the energy of the solution, we establish the existence of a regular global attractor $\A=\A(t)$. The kernel sections $\A(t)$ have finite fractal dimension.