Equi-attraction and continuity of attractors for skew-product semiflows
Tomás Caraballo Alexandre N. Carvalho Henrique B. da Costa José A. Langa
In this paper we prove the equivalence between equi-attraction and continuity of attractors for skew-product semi-flows, and equi-attraction and continuity of uniform and cocycle attractors associated to non-autonomous dynamical systems. To this aim proper notions of equi-attraction have to be introduced in phase spaces where the driving systems depend on a parameter. Results on the upper and lower-semicontinuity of uniform and cocycle attractors are relatively new in the literature, as a deep understanding of the internal structure of these sets is needed, which is generically difficult to obtain. The notion of lifted invariance for uniform attractors allows us to compare the three types of attractors and introduce a common framework in which to study equi-attraction and continuity of attractors. We also include some results on the rate of attraction to the associated attractors.
keywords: skew-product semiflows. Equi-attraction continuity of attractors
Strongly damped wave equations in $W^(1,p)_0 (\Omega) \times L^p(\Omega)$
Alexandre N. Carvalho Jan W. Cholewa
This paper is devoted to well posedness and regularity of the solutions of

$u_t_t + 2\nA^(1/2)u_t + A\u = \f(u)$

in $W^(1,p)_0 (\Omega) \times L^p(\Omega), p \in (1,\infty)$, where $\Omega \subset \mathbb {R}^N$ is a bounded smooth domain, $\n$ > 0 and -$\A$ is the Dirichlet Laplacian in $L^p(\Omega)$. We prove local well posedness result for nonlinearities $\f : \mathbb {R} \rightarrow \mathbb {R}$ satisfying $|f(s) - f(t)| \<= C|s - t|(1 + |s|^(p - 1) + |t|^(p - 1))$ with $\p < (N+p)/(N - p) (N > p)$, and show that the solutions are classical. If $f$ is dissipative and $p < (N+2)/(N - 2) (N \>= 3)$, we show that the associated semigroup has a global attractor $\cc{A}_(n,p)$ in $W^(1,p)_0(\Omega)\times L^p(\Omega)$, $p \in [2,\infty)$, which coincides with the attractor $\bb{A}_(n,2) =: \bb{A}_n$. We also obtain that $\bb{A}_n$ is compact in $C^(2+\mu)(bar(\Omega)) \times C^(1+\mu)(bar(\Omega))$ and attracts bounded subsets of $H^1_0(\Omega) \times L^2(\Omega)$ in $C^(2+\mu)(bar(\Omega)) \times C^(1+\mu)(bar(\Omega))$ for each $\mu \in (0, 1)$.

keywords: Analytic semigroup well posedness global attractor regularity.
NLS-like equations in bounded domains: Parabolic approximation procedure
Alexandre N. Carvalho Jan W. Cholewa

The article is devoted to semilinear Schrödinger equations in bounded domains. A unified semigroup approach is applied following a concept of Trotter-Kato approximations.Critical exponents are exhibited and global solutions are constructed for nonlinearities satisfying even a certain critical growth condition in $ H^1_0(Ω)$.

keywords: Schrödinger equation parabolic equation critical exponents
Alexandre N. Carvalho José A. Langa James C. Robinson
We were very pleased to be given the opportunity by Prof. Peter Kloeden to edit this special issue of Discrete and Continuous Dynamical Systems - Series B on the asymptotic dynamics of non-autonomous systems.

For more information please click the “Full Text” above.
Non-autonomous dynamical systems
Alexandre N. Carvalho José A. Langa James C. Robinson
This review paper treats the dynamics of non-autonomous dynamical systems. To study the forwards asymptotic behaviour of a non-autonomous differential equation we need to analyse the asymptotic configurations of the non-autonomous terms present in the equations. This fact leads to the definition of concepts such as skew-products and cocycles and their associated global, uniform, and cocycle attractors. All of them are closely related to the study of the pullback asymptotic limits of the dynamical system, from which naturally emerges the concept of a pullback attractor. In the first part of this paper we want to clarify these different dynamical scenarios and the relations between their corresponding attractors.
    If the global attractor of an autonomous dynamical system is given as the union of a finite number of unstable manifolds of equilibria, a detailed understanding of the continuity of the local dynamics under perturbation leads to important results on the lower-semicontinuity and topological structural stability for the pullback attractors of evolution processes that arise from small non-autonomous perturbations, with respect to the limit regime. Finally, continuity with respect to global dynamics under non-autonomous perturbation is also studied, for which appropriate concepts for Morse decomposition of attractors and non-autonomous Morse--Smale systems are introduced. All of these results will also be considered for uniform attractors. As a consequence, this paper also makes connections between different approaches to the qualitative theory of non-autonomous differential equations, which are often treated independently.
keywords: Morse decomposition pullback attractor Skew-product semiflow uniform attractor. cocycle attractor Morse--Smale systems

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