José A. Langa

Attractors for infinite-dimensional non-autonomous dynamical systems /

The pullback attractor.- Existence results for pullback attractors.- Continuity of attractors.- Finite-dimensional attractors.- Gradient semigroups and their dynamical properties.- Semilinear Differential Equations.- Exponential dichotomies.- Hyperbolic solutions and their stable and unstable manifolds.- A non-autonomous competitive Lotka-Volterra system.- Delay differential equations.-The Navier-Stokes equations with non-autonomous forcing.- Applications to parabolic problems.- A non-autonomous Chafee-Infante equation.- Perturbation of diffusion and continuity of attractors with rate.- A non-autonomous damped wave equation.- References.- Index.-

Flattening, squeezing and the existence of random attractors

The study of qualitative properties of random and stochastic differential equations is now one of the most active fields in the modern theory of dynamical systems. In the deterministic case, the properties of flattening and squeezing in infinite-dimensional autonomous dynamical systems require the existence of a bounded absorbing set and imply the existence of a global attractor. The flattening property involves the behaviour of individual trajectories while the squeezing property involves the difference of trajectories. It is shown here that the flattening property is implied by the squeezing property and is in fact weaker, since the attractor in a system with the flattening property can be infinite-dimensional, whereas it is always finite-dimensional in a system with the squeezing property. The flattening property is then generalized to random dynamical systems, for which it is called the pullback flattening property. It is shown to be weaker than the random squeezing property, but equivalent to pullback asymptotic compactness and pullback limit-set compactness, and thus implies the existence of a random attractor. The results are also valid for deterministic non-autonomous dynamical systems formulated as skew-product flows.

Pullback Attractors of Nonautonomous and Stochastic Multivalued Dynamical Systems

In this paper we study the existence of pullback global attractors for multivalued processes generated by differential inclusions. First, we define multivalued dynamical processes, prove abstract results on the existence of ω-limit sets and global attractors, and study their topological properties (compactness, connectedness). Further, we apply the abstract results to nonautonomous differential inclusions of the reaction–diffusion type in which the forcing term can grow polynomially in time, and to stochastic differential inclusions as well.

A Stochastic Pitchfork Bifurcation in a Reaction-Diffusion Equation

We study in some detail the structure of the random attractor for the Chafee-Infante reaction-diffusion equation perturbed by a multiplicative white noise, du=( Δu+βu- u 3 ) dt+σuod W t ,x∈D⊂ R m First we prove, for m ⩽ 5, a lower bound on the dimension of the random attractor, which is of the same order in β as the upper bound we derived in an earlier paper, and is the same as that obtained in the deterministic case. Then we show, for m = 1, that as β passes through λ1 (the first eigenvalue of the negative Laplacian) from below, the system undergoes a stochastic bifurcation of pitchfork type. We believe that this is the first example of such a stochastic bifurcation in an infinite–dimensional setting. Central to our approach is the existence of a random unstable manifold.

Stability, instability, and bifurcation phenomena in non-autonomous differential equations

There is a vast body of literature devoted to the study of bifurcation phenomena in autonomous systems of differential equations. However, there is currently no well-developed theory that treats similar questions for the non-autonomous case. Inspired in part by the theory of pullback attractors, we discuss generalizations of various autonomous concepts of stability, instability, and invariance. Then, by means of relatively simple examples, we illustrate how the idea of a bifurcation as a change in the structure and stability of invariant sets remains a fruitful concept in the non-autonomous case.

Existence of pullback attractors for pullback asymptotically compact processes

Abstract This paper is concerned with the existence of pullback attractors for evolution processes. Our aim is to provide results that extend the following results for autonomous evolution processes (semigroups) (i) An autonomous evolution process which is bounded, dissipative and asymptotically compact has a global attractor. (ii) An autonomous evolution process which is bounded, point dissipative and asymptotically compact has a global attractor. The extension of such results requires the introduction of new concepts and brings up some important differences between the asymptotic properties of autonomous and non-autonomous evolution processes. An application to damped wave problem with non-autonomous damping is considered.

FINITE DIMENSIONALITY OF ATTRACTORS FOR NON-AUTONOMOUS DYNAMICAL SYSTEMS GIVEN BY PARTIAL DIFFERENTIAL EQUATIONS

In the last decade, the concept of pullback attractor has become one of the usual tools to describe some qualitative properties of non-autonomous partial differential equations. A pullback attractor is a family of compact sets, invariant for the corresponding process related to the equation, and attracting from the past, and it assumes a natural generalization of the now classical concept of global attractor for autonomous partial differential equations. In this work we give sufficient conditions in order to prove the finite Hausdorff and fractal dimensionality of pullback attractors for non-autonomous infinite dimensional dynamical systems, and we apply our results to a generalized non-autonomous partial differential equation of Navier–Stokes type.

The effect of noise on the Chafee-Infante equation : a nonlinear case study

We investigate the effect of perturbing the Chafee-Infante scalar reaction diffusion equation, u(t) - Delta u = beta u- u(3), by noise. While a single multiplicative Ito noise of sufficient intensity will stabilise the origin, its Stratonovich counterpart leaves the dimension of the attractor essentially unchanged. We then show that a collection of multiplicative Stratonovich terms can make the origin exponentially stable, while an additive noise of sufficient richness reduces the random attractor to a single point.

Permanence and asymptotically stable complete trajectories for nonautonomous Lotka-Volterra models with diffusion

Lotka–Volterra systems are the canonical ecological models used to analyze population dynamics of competition, symbiosis, or prey-predator behavior involving different interacting species in a fixed habitat. Much of the work on these models has been within the framework of infinite-dimensional dynamical systems, but this has frequently been extended to allow explicit time dependence, generally in a periodic, quasiperiodic, or almost periodic fashion. The presence of more general nonautonomous terms in the equations leads to nontrivial difficulties which have stalled the development of the theory in this direction. However, the theory of nonautonomous dynamical systems has received much attention in the last decade, and this has opened new possibilities in the analysis of classical models with general nonautonomous terms. In this paper we use the recent theory of attractors for nonautonomous PDEs to obtain new results on the permanence and the existence of forwards and pullback asymptotically stable global...

Stability of Gradient Semigroups Under Perturbations

In this paper we prove that gradient-like semigroups (in the sense of Carvalho and Langa (2009 J. Diff. Eqns 246 2646–68)) are gradient semigroups (possess a Lyapunov function). This is primarily done to provide conditions under which gradient semigroups, in a general metric space, are stable under perturbation exploiting the known fact (see Carvalho and Langa (2009 J. Diff. Eqns 246 2646–68)) that gradient-like semigroups are stable under perturbation. The results presented here were motivated by the work carried out in Conley (1978 Isolated Invariant Sets and the Morse Index (CBMS Regional Conference Series in Mathematics vol 38) (RI: American Mathematical Society Providence)) for groups in compact metric spaces (see also Rybakowski (1987 The Homotopy Index and Partial Differential Equations (Universitext) (Berlin: Springer)) for the Morse decomposition of an invariant set for a semigroup on a compact metric space).