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In this work we define the generalizedφ-pullback attractors for evolution processes in complete metric spaces, which are compact and positively invariant families, that pullback attract bounded sets with a rate determined by a decreasing function φ that vanishes at infinity, called decay function. We find conditions under which a given evolution process has a generalized φ-pullback attractor, both in the discrete and in the continuous cases. We present a result for the special case of generalized polynomial pullback attractors, and apply it to obtain such an object for a nonautonomous wave equation.

This paper is concerned with the pullback measure attractors of the non-autonomous fractional reaction-diffusion equations defined on Rn. We first prove the existence and uniqueness of pullback measure attractors for such equations. Then we establish the upper semi-continuity of these attractors as the noise intensity ε tends to zero. Specifically, we apply the uniform estimates on the tails of solutions to prove the asymptotic compactness of a family of probability distributions of solutions to overcome the non-compactness of usual Sobolev embeddings on unbounded domains.

For a smooth quasi-projective surface S over C we consider the Borel–Moore homology of the stack of coherent sheaves on S with compact support and make this space into an associative algebra by a version of the Hall multiplication. This multiplication involves data (virtual pullbacks) governing the derived moduli stack, i.e., the perfect obstruction theory naturally existing on the non-derived stack. By restricting to sheaves with support of given dimension, we obtain several types of Hecke operators. In particular, we study R(S) , the Hecke algebra of 0 -dimensional sheaves. For the case S=A^2 , we show that R(S) is an enveloping algebra and identify it, as a vector space, with the symmetric algebra of an explicit graded vector space. For a general S , we find the graded dimension of R(S) , using the techniques of factorization cohomology.

This article is dedicated to investigating limit behaviours of invariant measures with respect to delay and system parameters of 3D Navier–Stokes–Voigt equations. Firstly, the well-posedness of such a system is obtained on arbitrary open sets that satisfy the Poincaré inequality, and then a unique minimal pullback attractor is attained by using the energy equation method and asymptotic compactness property. Furthermore, we construct a family of invariant Borel probability measures, which are supported on the pullback attractors. Specifically, when the external forcing terms are periodic in time, the periodic invariant measure can be obtained. Finally, as the delay approaches zero and system parameters tend to some numbers, the limit of the invariant measure sequences for this class of equations must be the invariant measure of the corresponding limit equations.

The aim of this article is to study the asymptotic behaviour of non-autonomous stochastic lattice systems. We first show the existence and uniqueness of a pullback measure attractor. Moreover, when deterministic external forcing terms are periodic in time, we show the pullback measure attractors are periodic. We then study the upper semicontinuity of pullback measure attractors as the noise intensity goes to zero. Pullback asymptotic compact for a family of probability measures with respect to probability distributions of the solutions is demonstrated by using uniform a priori estimates for far-field values of solutions.

This paper is concerned with the asymptotic behaviour of solutions to a class of non-autonomous stochastic nonlinear wave equations with dispersive and viscosity dissipative terms driven by operator-type noise defined on the entire space $\\mathbb {R}^n$. The existence, uniqueness, time-semi-uniform compactness and asymptotically autonomous robustness of pullback random attractors are proved in $H^1(\\mathbb {R}^n)\\times H^1(\\mathbb {R}^n)$ when the growth rate of the nonlinearity has a subcritical range, the density of the noise is suitably controllable, and the time-dependent force converges to a time-independent function in some sense. The main difficulty to establish the time-semi-uniform pullback asymptotic compactness of the solutions in $H^1(\\mathbb {R}^n)\\times H^1(\\mathbb {R}^n)$ is caused by the lack of compact Sobolev embeddings on $\\mathbb {R}^n$, as well as the weak dissipativeness of the equations is surmounted at light of the idea of uniform tail-estimates and a spectral decomposition approach. The measurability of random attractors is proved by using an argument which considers two attracting universes developed by Wang and Li (Phys. D 382: 46–57, 2018).

Our purpose is to study some continuity properties of pullback and pullback exponential attractors for the non-autonomous plate with p-Laplacian and nonlocal weak damping gϵ(‖ut‖)ut under hinged boundary condition. Moreover, the existence of pullback attractors in the natural space energy with finite dimensionality is proved together with its upper semicontinuity and continuity with respect to the perturbed parameter ϵ∈[0,1]. Finally, we prove that the related process has a pullback exponential attractor Mexpϵ and is Hölder continuous on ϵ∈[0,1]. In particular, the continuity on perturbation ϵ∈[0,1] holds for global and exponential attractors when the non-autonomous dynamical system degenerates to an autonomous one.

We consider the global well-posedness of a non-autonomous Klein–Gordon–Schrödinger type system that models a scalar nucleons interacting with neutral mesons in tree spatial dimension. Moreover, we establish the existence of the weak pullback attractor as well the existence of the strong pullback attractor is obtained in a more regular space.

The paper investigates the global well-posedness and the longtime dynamics for a class of strongly damped wave equations with evolutional p(x, t)-Laplacian and q(x, t)-growth source term on a bounded domain Ω⊂R3:utt-∇·(|∇u|p(x,t)-2∇u)-λΔu-Δut+|u|q(x,t)-2u=g, together with the perturbed parameter λ∈[0,1] and the Dirichlet boundary condition. We show that under rather relaxed conditions, (i) the model is global well-posed; (ii) for each λ0∈(0,1], the related nonautonomous dynamical systems acting on the time-dependent phase spaces have a family of pullback D-exponential attractor Eλ=Eλ(t)t∈R∈D which is Hölder continuous w.r.t. λ at λ0; (iii) they have also a family of finite dimensional pullback D-attractors Aλ=Aλ(t)t∈R which are upper semicontinuous and residual continuous w.r.t. λ∈(0,1]. In particular, when λ∈(0,1] and without the p(x, t)-Laplacian, the above mentioned results can be greatly improved, in the concrete; (iv) the weak solutions of the corresponding model possess additionally partial regularity and the Hölder stability in stronger H1×H1-norm, the pullback D-attractor and pullback D-exponential attractor in weaker Y1-norm can be regularized to be those in stronger H1×H1-norm, which are also the standard ones in Ht-norm. The method provided here allows overcoming the difficulties arising from variable exponent nonlinearities and extending the analysis and the results for these type of models with constant exponent nonlinearities.