Strong Uniform Attractors for Non-Autonomous Dissipative PDEs with non translation-compact external forces

We give a comprehensive study of strong uniform attractors of non-autonomous dissipative systems for the case where the external forces are not translation compact. We introduce several new classes of external forces which are not translation compact, but nevertheless allow to verify the attraction in a strong topology of the phase space and discuss in a more detailed way the class of so-called normal external forces introduced before. We also develop a unified approach to verify the asymptotic compactness for such systems based on the energy method and apply it to a number of equations of mathematical physics including the Navier-Stokes equations, damped wave equations and reaction-diffusing equations in unbounded domains

Averaging of equations of viscoelasticity with singularly oscillating external forces

Given $\rho\in[0,1]$, we consider for $\varepsilon\in(0,1]$ the nonautonomous viscoelastic equation with a singularly oscillating external force $$\partial_{tt} u-\kappa(0)\Delta u - \int_0^\infty \kappa'(s)\Delta u(t-s) d s +f(u)=g_{0}(t)+\varepsilon ^{-\rho }g_{1}(t/\varepsilon )$$ together with the {\it averaged} equation $$\partial_{tt} u-\kappa(0)\Delta u - \int_0^\infty \kappa'(s)\Delta u(t-s) d s +f(u)=g_{0}(t).$$ Under suitable assumptions on the nonlinearity and on the external force, the related solution processes $S_\varepsilon(t,\tau)$ acting on the natural weak energy space ${\mathcal H}$ are shown to possess uniform attractors ${\mathcal A}^\varepsilon$. Within the further assumption $\rho<1$, the family ${\mathcal A}^\varepsilon$ turns out to be bounded in ${\mathcal H}$, uniformly with respect to $\varepsilon\in[0,1]$. The convergence of the attractors ${\mathcal A}^\varepsilon$ to the attractor ${\mathcal A}^0$ of the averaged equation as $\varepsilon\to 0$ is also established

Evolutionary system, global attractor, trajectory attractor and applications to the nonautonomous reaction-diffusion systems

In [Adv. Math., 267(2014), 277-306], Cheskidov and Lu develop a new framework of the evolutionary system that deals directly with the notion of a uniform global attractor due to Haraux, and by which a trajectory attractor is able to be defined for the original system under consideration. The notion of a trajectory attractor was previously established for a system without uniqueness by considering a family of auxiliary systems including the original one. In this paper, we further prove the existence of a notion of a strongly compact strong trajectory attractor if the system is asymptotically compact. As a consequence, we obtain the strong equicontinuity of all complete trajectories on global attractor and the finite strong uniform tracking property. Then we apply the theory to a general nonautonomous reaction-diffusion systems. In particular, we obtain the structure of uniform global attractors without any additional condition on nonlinearity other than those guarantee the existence of a uniform absorbing set. Finally, we construct some interesting examples of such nonlinearities. It is not known whether they can be handled by previous frameworks.Comment: A new paper arXiv:1811.05783 is submitted, which includes the subjects in this paper, but much more. It is not an updated one, it is completely rewritten and is much longer and cleare

The Existence of Exponential Attractor for Discrete Ginzburg-Landau Equation

This paper studies the following discrete systems of the complex Ginzburg-Landau equation: iu˙m-(α-iε)(2um-um+1-um-1)+iκum+βum2σum=gm,  m∈Z. Under some conditions on the parameters α, ε, κ, β, and σ, we prove the existence of exponential attractor for the semigroup associated with these discrete systems

Pullback Exponential Attractors for Nonautonomous Klein-Gordon-Schrödinger Equations on Infinite Lattices

This paper proves the existence of the pullback exponential attractor for the process associated to the nonautonomous Klein-Gordon-Schrödinger equations on infinite lattices

Asymptotic behaviour of nonlocal p-Laplacian reaction-diffusion problems

In this paper, we focus on studying the existence of attractors in the phase spaces L2(Ω) and Lp(Ω) (among others) for time-dependent p-Laplacian equations with nonlocal diffusion and nonlinearities of reaction-diffusion type. Firstly, we prove the existence of weak solutions making use of a change of variable which allows us to get rid of the nonlocal operator in the diffusion term. Thereupon, the regularising effect of the equation is shown applying an argument of a posteriori regularity, since under the assumptions made we cannot guarantee the uniqueness of weak solutions. In addition, this argument allows to ensure the existence of an absorbing family in W1,p 0 (Ω). This leads to the existence of the minimal pullback attractors in L2(Ω), Lp(Ω) and some other spaces as Lp∗−(Ω). Relationships between these families are also established.Ministerio de Economía y CompetitividadFondo Europeo de Desarrollo RegionalJunta de Andalucí