Attractors for Parametric Delay Differential Equations Without Uniqueness and Their Upper Semicontinuous Behaviour

We prove existence of a global attractor A(λ) under minimal assumptions for a general class of parameterized delay differential equations without uniqueness and posed in potentially different state spaces. Secondly, we establish the upper semicontinuity of the attractors with respect to the parameter λ

Attractors for Parametric Delay Differential Equations and Their Continuous Behavior

The problem of the continuity of global attractors under minimal assumptions for a general class of parameterized delay differential equations is considered. The theory of equi-attraction developed by Li and Kloeden in [2004a] is adapted to this framework, so it is proved that the continuity of the attractors with respect to the parameter is equivalent to this equi-attraction property

Stability results for 2D Navier-Stokes equations with unbounded delay

Some results related to 2D Navier-Stokes equations when the external force contains hereditary characteristics involving unbounded delays are analyzed. First, the existence and uniqueness of solutions is proved by Galerkin approximations and the energy method. The existence of stationary solution is then established by means of the Lax-Milgram theorem and the Schauder fixed point theorem. The local stability analysis of stationary solutions is studied by several different methods: the classical Lyapunov function method, the Razumikhin-Lyapunov technique and by constructing appropriate Lyapunov functionals. Finally, we also verify the polynomial stability of the stationary solution in a particular case of unbounded variable delay. Exponential stability in this infinite delay setting remains as an open problem.Ministerio de Economía y CompetitividadFondo Europeo de Desarrollo RegionalJunta de AndalucíaNational Science Foundation of ChinaScience and Technology Commission of Shanghai MunicipalityShanghai Leading Academic Discipline Projec

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

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

Global Attractor and Omega-Limit Sets Structure for a Phase-Field Model of Thermal Alloys

In this paper, the existence of weak solutions is established for a phase-field model of thermal alloys supplemented with Dirichlet boundary conditions. After that, the existence of global attractors for the associated multi-valued dynamical systems is proved, and the relationship among these sets is established. Finally, we provide a more detailed description of the asymptotic behaviour of solutions via the omega-limit sets. Namely, we obtain a characterization–through the natural stationary system associated to the model–of the elements belonging to the omega-limit sets under suitable assumptions

Pullback exponential attractors for parabolic equations with dynamical boundary conditions

The existence of pullback exponential attractors for a nonautonomous semilinear parabolic equation with dynamical boundary condition is proved when the timedependent forcing terms are translation bounded or even grow exponentially in the past and in the future.Ministerio de Educación: Dirección General de Política Universitari

Dynamics of wave equations with moving boundary

This paper is concerned with long-time dynamics of weakly damped semilinear wave equations defined on domains with moving boundary. Since the boundary is a function of the time variable the problem is intrinsically non-autonomous. Under the hypothesis that the lateral boundary is time-like, the solution operator of the problem generates an evolution process U(t, τ ) : Xτ → Xt, where Xt are timedependent Sobolev spaces. Then, by assuming the domains are expanding, we establish the existence of minimal pullback attractors with respect to a universe of tempered sets defined by the forcing terms. Our assumptions allow nonlinear perturbations with critical growth and unbounded time-dependent external forces.Conselho Nacional de Desenvolvimento Científico e TecnológicoMinisterio de EducaciónMinisterio de Ciencia e Innovació

Equi-atracción y dependencia continua de atractores para ecuaciones con retardo

En esta comunicación se presentan brevemente los resultados del trabajo [P. E. Kloeden & P. Marín-Rubio. Equi-Attraction and the continuous dependence of attractors on time delays. Discrete Contin. Dyn. Syst. Ser. A.]: condiciones de equivalencia sobre la continuidad de atractores para semiflujos generados por ecuaciones diferenciales con retardo planteadas en (potencialmente) distintos espacios de fase. Dichas condiciones, basadas en el concepto de equi-atracción, son una extensión de trabajos previos debidos a Li y Kloeden (e.g. [D.S. Li & P.E. Kloeden, Equi-attraction and the continuous dependence of attractors on parameters, Glasgow Math. J., 46 (2004), 131–141]), y requieren previamente la inmersión de todos los sistemas dinámicos en un marco que los englobe, y la reinterpretación correcta de las condiciones entre los espacios originales y éste último. Se añaden ejemplos que ilustran la teoría.Ministerio de Educación y CienciaFondo Europeo de Desarrollo Regiona

Some Results on Stochastic Differential Equations with Reflecting Boundary Conditions

Some results related to stochastic differential equations with reflecting boundary conditions (SDER) are obtained. Existence and uniqueness of strong solution is ensured under the relaxation on the drift coefficient (instead of the Lipschitz character, a monotonicity condition is supposed)