Time-dependent attractors for non-autonomous nonlocal reaction-diffusion equations

In this paper, the existence and uniqueness of weak and strong solutions for a non-autonomous nonlocal reaction-diffusion equation is proved. Next, the existence of minimal pullback attractors in the L2 -norm in the frameworks of universes of fixed bounded sets and those given by a tempered growth condition, and some relationships between them are established. Finally, we prove the existence of minimal pullback attractors in the H1-norm and study relationships among these new families and those given previously in the L2 - context. The results are also new in the autonomous framework in order to ensure the existence of global compact attractors, as a particular case.Ministerio de Economía y CompetitividadFondo Europeo de Desarrollo RegionalJunta de Andalucí

Sufficient and Necessary Criteria for Existence of Pullback Attractors for Non-compact Random Dynamical Systems

We study pullback attractors of non-autonomous non-compact dynamical systems generated by differential equations with non-autonomous deterministic as well as stochastic forcing terms. We first introduce the concepts of pullback attractors and asymptotic compactness for such systems. We then prove a sufficient and necessary condition for existence of pullback attractors. We also introduce the concept of complete orbits for this sort of systems and use these special solutions to characterize the structures of pullback attractors. For random systems containing periodic deterministic forcing terms, we show the pullback attractors are also periodic. As an application of the abstract theory, we prove the existence of a unique pullback attractor for Reaction-Diffusion equations on $\R^n$ with both deterministic and random external terms. Since Sobolev embeddings are not compact on unbounded domains, the uniform estimates on the tails of solutions are employed to establish the asymptotic compactness of solutions.Comment: References adde

Robustness of time-dependent attractors in H1-norm for nonlocal problems

In this paper, the existence of regular pullback attractors as well as their upper semicontinuous behaviour in H1-norm are analysed for a parameterized family of non-autonomous nonlocal reaction-diffusion equations without uniqueness, improving previous results [Nonlinear Dyn. 84 (2016), 35–50].Ministerio de Economía y CompetitividadFondo Europeo de Desarrollo RegionalJunta de Andalucí

Pullback attractors for reaction-diffusion equations in some unbounded domains with an H-1 -valued non-autonomous forcing term and without uniqueness of solutions

The existence of a pullback attractor for a reaction-diffusion equations in an unbounded domain containing a non-autonomous forcing term taking values in the space H−1, and with a continuous nonlinearity which does not ensure uniqueness of solutions, is proved in this paper. The theory of set-valued non-autonomous dynamical systems is applied to the problem