Upper Semicontinuity Of Pullback Attractors Research Articles

Multivalued dynamics of non-autonomous reaction–diffusion equation with nonlinear advection term

In this paper, we investigate a reaction–diffusion population model with a nonlinear advection term and a time-dependent force given by the equation ut−Δu+α→⋅∇up=f(u)+h(t)in(τ,∞)×Ω,subject to the boundary condition u=0 on (τ,∞)×∂Ω. Here, Ω⊂RN with N≥1 is a bounded domain with smooth boundary, τ∈R, α→=(α1,…,αN) is a given advective direction and p>1. The presence of the nonlinear advection term α→⋅∇up introduces technical difficulties in the analysis, leading to a scenario where the uniqueness of weak solutions cannot be guaranteed. Consequently, the equation generates a multi-valued nonautonomous dynamical system. In this context, we establish the existence of minimal pullback attractors, considering universes of bounded and tempered sets. Moreover, we explore the relationships between these pullback attractors. Finally, we prove the upper semicontinuity of pullback attractors with respect to the advective vector α→.

Existence and upper semicontinuity of pullback attractors for Kirchhoff wave equations in time-dependent spaces

Existence and upper semicontinuity of pullback attractors for Kirchhoff wave equations in time-dependent spaces

Long-time behavior of nonclassical diffusion equations with memory on time-dependent spaces

This paper aims to study the long-time behavior of nonclassical diffusion equation with memory and disturbance parameters on time-dependent space. By using the contractive process method on the family of time-dependent spaces and operator decomposition technique, the existence of pullback attractors is first proved. Then the upper semi-continuity of pullback attractors with respect to perturbation parameter ν in M t is obtained. It’s remarkable that the nonlinearity f satisfies the polynomial growth of arbitrary order.

Upper semicontinuity of pullback attractors for nonclassical diffusion equations with delay

In this paper, we mainly study the upper semicontinuity of pullback D-attractors for a nonclassical diffusion equation with delay term b ( t , u t ) which contains some hereditary characteristics. Under a critical nonlinearity f, a time-dependent force g ( t , x ) with exponential growth and a delayed force term b ( t , u t ), using the asymptotic a priori estimate method, we prove the upper semicontinuity of pullback D-attractor { A ε ( t ) } t ∈ R to equation (1.1) with ε ∈ [ 0 , 1 ].

Upper semi-continuity of pullback attractors for bipolar fluids with delay

<abstract><p>We investigate bipolar fluids with delay in a $ 2D $ channel $ \Sigma = \mathbb{R}\times (-K, K) $ for some $ K > 0 $. The channel $ \Sigma $ is divided into a sequence of simply connected, bounded, and smooth sub-domains $ \Sigma_n (n = 1, 2, 3\cdot\cdot\cdot) $, such that $ \Sigma_n\rightarrow \Sigma $ as $ n\rightarrow \infty $. The paper demonstrates that the pullback attractors in the sub-domains $ \Sigma_n $ converge to the pullback attractor in the entire domain $ \Sigma $ as $ n\rightarrow \infty. $</p></abstract>

Regularity and upper semicontinuity of pullback attractors for non-autonomous Rao–Nakra beam

In this paper we study the long-time dynamics of a non-autonomous Rao–Nakra sandwich beam. The governing equations of Rao–Nakra sandwich beam consist of two wave equations for the longitudinal displacements of the top and bottom layers, and one Euler–Bernoulli beam equation for the transversal displacement. Under quite general assumptions on nonlinear damping and sources terms and based on nonlinear semigroups and the theory of monotone operators, we establish existence and uniqueness of weak and strong solutions. We also establish a Lipschitz stability result. We prove the existence of pullback attractors in natural space energy. Finally, we prove the regularity of the family of pullback attractors and its upper semicontinuous with respect to the fractional exponent γ ∈ (0, 1/2).

Well-posedness and dynamics of double time-delayed lattice FitzHugh–Nagumo systems

This paper is concerned with the well-posedness and dynamics of FitzHugh–Nagumo systems with variable delay defined on infinite lattices. We first prove the existence and uniqueness of solutions as well as the existence of an evolution process for this system. We then establish the existence of tempered pullback attractors and invariant measures. Finally, we study the upper semicontinuity of pullback attractors as the delay time tends to zero.

Upper semicontinuity of pullback attractors for non-autonomous lattice systems under singular perturbations

<p style='text-indent:20px;'>Consider the second order nonautonomous lattice systemswith singular perturbations</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \epsilon \ddot{u}_{m}+\dot{u}_{m}+(Au)_{m}+\lambda_{m}u_{m}+f_{m}(u_{j}|j\in I_{mq}) = g_{m}(t),\; \; m\in \mathbb{Z}^{k},\; \; \epsilon>0 \tag{*} \label{0} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>and the first order nonautonomous lattice systems</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \begin{equation*} \dot{u}_{m}+(Au)_{m}+\lambda _{m}u_{m}+f_{m}(u_{j}|j∈I_{mq}) = g_{m}(t),\; \; m\in \mathbb{Z}^{k}. \tag{**} \label{00} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>Under certain conditions, there are pullback attractors <inline-formula><tex-math id="M1">\begin{document}$ \{\mathcal{A}_{\epsilon }(t)\subset \ell ^{2}\times \ell ^{2}\}_{t\in \mathbb{R}} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \{\mathcal{A}(t)\subset \ell ^{2}\}_{t\in \mathbb{R}} $\end{document}</tex-math></inline-formula> for systems (*)and (**), respectively. In this paper, we mainly consider the uppersemicontinuity of attractors <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{A}_{\epsilon }(t)\subset \ell^{2}\times \ell ^{2} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ t\in \mathbb{R} $\end{document}</tex-math></inline-formula>, with respect to the coefficient <inline-formula><tex-math id="M5">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> of second derivative term under Hausdorff semidistance. First, we studythe relationship between <inline-formula><tex-math id="M6">\begin{document}$ \mathcal{A}_{\epsilon }(t) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ \mathcal{A}(t) $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M8">\begin{document}$ \epsilon \rightarrow 0^{+} $\end{document}</tex-math></inline-formula>. We construct a family of compact sets <inline-formula><tex-math id="M9">\begin{document}$ \mathcal{A}_{0}(t)\subset \ell ^{2}\times \ell ^{2} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ t\in \mathbb{R} $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M11">\begin{document}$ \mathcal{A}(t) $\end{document}</tex-math></inline-formula> is naturally embedded into <inline-formula><tex-math id="M12">\begin{document}$ \mathcal{A}_{0}(t) $\end{document}</tex-math></inline-formula> as the firstcomponent, and prove that <inline-formula><tex-math id="M13">\begin{document}$ \mathcal{A}_{\epsilon }(t) $\end{document}</tex-math></inline-formula> can enter anyneighborhood of <inline-formula><tex-math id="M14">\begin{document}$ \mathcal{A}_{0}(t) $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M15">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> is small enough. Thenfor <inline-formula><tex-math id="M16">\begin{document}$ \epsilon _{0}>0 $\end{document}</tex-math></inline-formula>, we prove that <inline-formula><tex-math id="M17">\begin{document}$ \mathcal{A}_{\epsilon }(t) $\end{document}</tex-math></inline-formula> can enterany neighborhood of <inline-formula><tex-math id="M18">\begin{document}$ \mathcal{A}_{\epsilon _{0}}(t) $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M19">\begin{document}$ \epsilon\rightarrow \epsilon _{0} $\end{document}</tex-math></inline-formula>. Finally, we consider the existence andexponentially attraction of the singleton pullback attractors of systems (*)-(**).</p>

Long-time behaviour for a non-autonomous Klein-Gordon-Zakharov system

The aim of this paper is to study the long-time dynamics of solutions of the evolution system{utt−Δu+u+η(−Δ)12ut+aϵ(t)(−Δ)12vt=f(u),(x,t)∈Ω×(τ,∞),vtt−Δv+η(−Δ)12vt−aϵ(t)(−Δ)12ut=0,(x,t)∈Ω×(τ,∞), subject to boundary conditionsu=v=0,(x,t)∈∂Ω×(τ,∞), where Ω is a bounded smooth domain in Rn, n≥3, with the boundary ∂Ω assumed to be regular enough, η>0 is constant, aϵ is a Hölder continuous function and f is a dissipative nonlinearity. This problem is a non-autonomous version of the well known Klein-Gordon-Zakharov system. Using the uniform sectorial operators theory, we will show the local and global well-posedness of this problem in H01(Ω)×L2(Ω)×H01(Ω)×L2(Ω). Additionally, we prove existence, regularity and upper semicontinuity of pullback attractors.

Upper semicontinuity of pullback attractors for a nonautonomous damped wave equation

In this paper, we study the local uniformly upper semicontinuity of pullback attractors for a strongly damped wave equation. In particular, under some proper assumptions, we prove that the pullback attractor {A_{varepsilon }(t)}_{tin mathbb{R}} of Eq. (1.1) with varepsilon in [0,1] satisfies lim_{varepsilon to varepsilon _{0}}sup_{tin [a,b]} operatorname{dist}_{H_{0}^{1}times L^{2}}(A_{varepsilon }(t),A_{ varepsilon _{0}}(t))=0 for any [a,b]subset mathbb{R} and varepsilon _{0}in [0,1].

High-order Wong-Zakai approximations for non-autonomous stochastic p -Laplacian equations on \mathbb{R}^N

In this paper, we investigate the approximations of stochastic \begin{document}$ p $\end{document} -Laplacian equation with additive white noise by a family of piecewise deterministic partial differential equations driven by a stationary stochastic process. We firstly obtain the tempered pullback attractors for the random \begin{document}$ p $\end{document} -Laplacian equation with a general diffusion. We secondly prove the convergence of solutions and the upper semi-continuity of pullback attractors of the Wong-Zakai approximation equations in a Hilbert space for the additive case. Thirdly, by a truncation technique, the uniform compactness of pullback attractor with respect to the quantity of approximations is derived in the space of \begin{document}$ q $\end{document} -times integrable functions, where the upper semi-continuity of the attractors of the approximation equations is well established.

Asymptotic behavior for a non-autonomous model of neural fields with variable external stimuli

In this work we consider the class of nonlocal non-autonomous evolution problems in a bounded smooth domain \(\Omega\) in \(\mathbb{R}^{N}\) $$\displaylines{ \partial_t u(t,x) =- a(t)u(t,x) + b(t) \int_{\mathbb{R}^N} J(x,y)f(t,u(t,y))\,dy -h +S(t,x),\quad t\geq\tau \cr u(\tau,x)=u_\tau(x), }$$ with u(t,x)= 0 for \(t\geq\tau\) and \(x \in\mathbb{R}^N\backslash\Omega\). Under appropriate assumptions we study the asymptotic behavior of the evolution process, generated by this problem in a suitable Banach space. We prove results on existence, uniqueness and smoothness of the solutions and on the existence of pullback attractor for the evolution process. We also prove a continuous dependence of the evolution process with respect to the external stimuli function present in the model. Furthermore, using the continuous dependence of the evolution process, we prove the upper semicontinuity of pullback attractors with respect to the external stimuli function. We finish this article with a small discussion about the model and about a biological interpretation of the result on the continuous dependence of neuronal activity with respect to the external stimuli function.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/92/abstr.html

Robustness of Attractors for Non-autonomous Kirchhoff Wave Models with Strong Nonlinear Damping

The paper investigates the robustness of pullback attractors and pullback exponential attractors of the non-autonomous Kirchhoff wave models with strong nonlinear damping: $$u_{tt}- (1+\epsilon \Vert \nabla u\Vert ^2)\Delta u-\sigma (\Vert \nabla u\Vert ^2) \Delta u_t+f(u)=g(x,t)$$, where $$\epsilon \in [0,1]$$ is an extensibility parameter. It shows that when the growth exponent p of the nonlinearity f(u) is up to the supercritical range: $$1\le p<p^{**}\equiv \frac{N+4}{(N-4)^+}$$, (i) the related process has a pullback attractor $${\mathscr {A}}_\epsilon $$ in natural energy space $${\mathcal {H}}=(H^1_0\cap L^{p+1})\times L^2$$ for each $$\epsilon $$, and it is upper semicontinuous on the perturbation $$\epsilon $$; (ii) the related process has a partially strong pullback exponential attractor $${\mathcal {M}}_\epsilon $$ for each $$\epsilon $$, and it is Holder continuous on $$\epsilon \in [0,1]$$. These results deepen and extend those in recent literatures (Chueshov in J Diff Equ 252:1229–1262, 2012; Ding et al. in Appl Math Lett 76:40–45, 2018; Wang and Zhong in Discrete Contin Dyn Syst 7:3189–3209, 2013). The main novelty of this paper is that it provides a new method to investigate the upper semicontinuity of pullback attractors and the stability of pullback exponential attractors in supercritical nonlinearity case.

Regularity and upper semicontinuity of pullback attractors for a class of nonautonomous thermoelastic plate systems

Regularity and upper semicontinuity of pullback attractors for a class of nonautonomous thermoelastic plate systems

Pullback dynamics of 3D Navier–Stokes equations with nonlinear viscosity

This paper is concerned with pullback dynamics of 3D Navier–Stokes equations with variable viscosity and subject to time-dependent external forces. Our main result establishes the existence of finite-dimensional pullback attractors in a general setting involving tempered universes. We also present a sufficient condition on the viscosity coefficients that guarantees the attractors are nontrivial. We end the paper by showing the upper semi-continuity of pullback attractors as the non-autonomous perturbation vanishes.

Weak upper semicontinuity of pullback attractors for nonautonomous reaction-diffusion equations

Weak upper semicontinuity of pullback attractors for nonautonomous reaction-diffusion equations

Existence and upper semi-continuity of pullback attractors of a p-Laplacian equation with delay

In this paper, we study the dynamics of a non-autonomous p-Laplacian equation with delay. We first prove the existence and uniqueness of solutions for this equation by using the Galerkin approximations. Then, applying the Sobolev compactness theorem, Ascoli-Azerla theorem, and energy method, we show the existence of pullback attractors for the associated process in space C([−h, 0]; Lp) as well as in space C([−h, 0]; L2). The upper semi-continuity result of the pullback attractors is also established here.

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 Ll(RN)×L2(RN) for any l ∈ (2,p]. The asymptotic compactness of the first component of the system in Lp(RN) 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.

Upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations

The paper investigates the upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations with structural damping: \begin{document}$ u_{tt}-M(\|\nabla u\|^2)\Delta u+(-\Delta)^\alpha u_t+f(u) = g(x,t) $\end{document} , where \begin{document}$ \alpha\in(1/2, 1) $\end{document} is said to be a dissipative index. It shows that when the nonlinearity \begin{document}$ f(u) $\end{document} is of supercritical growth \begin{document}$ p: 1 \leq p , the related evolution process has a pullback attractor for each \begin{document}$ \alpha\in(1/2, 1) $\end{document} , and the family of pullback attractors is upper semicontinuous with respect to \begin{document}$ \alpha $\end{document} . These results extend those in [ 27 ] for autonomous Kirchhoff wave models.

Existence, regularity and upper semicontinuity of pullback attractors for the evolution process associated to a neural field model

Existence, regularity and upper semicontinuity of pullback attractors for the evolution process associated to a neural field model