Strongly damped wave equation and its Yosida approximations

In this work we study the continuity for the family of global attractors of the equations $u_{tt}-\Delta u-\Delta u_t-\varepsilon \Delta u_{tt}=f(u)$ at $\varepsilon=0$ when $\Omega$ is a bounded smooth domain of $\mathbb{R}^n$, with $n\geq 3$, and the nonlinearity $f$ satisfies a subcritical growth condition. Also, we obtain an uniform bound for the fractal dimension of these global attractors

Generalized φ\varphi-pullback attractors for evolution processes and application to a nonautonomous wave equation

In this work we define the generalized $\varphi$-pullback attractors for evolution processes in complete metric spaces, which are compact and positively invariant families, such that they pullback attract bounded sets with a rate determined by a decreasing function $\varphi$ that vanishes at infinity. We find conditions under which a given evolution process has a generalized $\varphi$-pullback attractor, both in the discrete and in the continuous cases. We present a result for the special case of generalized polynomial pullback attractors, and apply it to obtain such an object for a nonautonomous wave equation.Comment: 32 page

Attractors for impulsive non-autonomous dynamical systems and their relations

In this work, we deal with several different notions of attractors that may appear in the impulsive non-autonomous case and we explore their relationships to obtain properties regarding the different scenarios of asymptotic dynamics, such as the cocycle attractor, the uniform attractor and the global attractor for the impulsive skew-product semiflow. Lastly, we illustrate our theory by exhibiting an example of a non-classical non-autonomous parabolic equation with subcritical nonlinearity and impulses.Fondo Europeo de Desarrollo RegionalMinisterio de Economía y CompetitividadConsejería de Innovación, Ciencia y Empresa (Junta de Andalucía)Fundação de Amparo à Pesquisa do Estado de São Paul

Impulsive surfaces on dynamical systems

This work is devoted to the construction of impulsive sets in Rn. In the literature, there are many examples of impulsive dynamical systems whose impulsive sets are chosen in an abstract way, and in this paper we present sufficient conditions to characterize impulsive sets in Rn which satisfy some “tube conditions” and ensure a good behavior of the flow. Moreover, we present some examples to illustrate the theoretical results.Fondo Europeo de Desarrollo RegionalMinisterio de Economía y CompetitividadConsejería de Innovación, Ciencia y Empresa (Junta de Andalucía)Fundação de Amparo à Pesquisa do Estado de São Paul

A survey on impulsive dynamical systems

In this survey we provide an introduction to the theory of impulsive dynamical systems in both the autonomous and nonautonomous cases. In the former, we will show two different approaches which have been proposed to analyze such kind of dynamical systems which can experience some abrupt changes (impulses) in their evolution. But, unlike the autonomous framework, the nonautonomous one is being developed right now and some progress is being obtained over the recent years. We will provide some results on how the theory of autonomous impulsive dynamical systems can be extended to cover such nonautonomous situations, which are more often to occur in the real world.Fundação de Amparo à Pesquisa do Estado de São PauloConselho Nacional de Desenvolvimento Científico e TecnológicoFondo Europeo de Desarrollo RegionalMinisterio de Economía y CompetitividadConsejería de Innovación, Ciencia y Empresa (Junta de Andalucía

Skew Product Semiflows and Morse Decomposition

This paper is devoted to the investigation of the dynamics of non-autonomous differential equations. The description of the asymptotic dynamics of non-autonomous equations lies on dynamical structures of some associated limiting non-autonomous - and autonomous - differential equations (one for each global solution in the attractor of the driving semigroup of the associated skew product semi-flow). In some cases, we have infinitely many limiting problems (in contrast with the autonomous - or asymptotically autonomous - case for which we have only one limiting problem; that is, the semigroup itself). We concentrate our attention in the study of the Morse decomposition of attractors for these non-autonomous limiting problems as a mean to understand some of the asymptotics of our non-autonomous differential equations. In particular, we derive a Morse decomposition for the global attractors of skew product semiflows (and thus for pullback attractors of non-autonomous differential equations) from a Morse decomposition of the attractor for the associated driving semigroup. Our theory is well suited to describe the asymptotic dynamics of non-autonomous differential equations defined on the whole line or just for positive times, or for differential equations driven by a general semigroup

[Book of abstracts]

USPFAPESPCAPESICMC Summer Meeting on Differential Equations (2015 São Carlos

Book of Abstracts

USPCAPESFAPESPCNPqINCTMatICMC Summer Meeting on Differentail Equations.\ud São Carlos, Brasil. 3-7 february 2014

Estrutura de atratores e estimativas de suas dimensões fractais

This work is dedicated to the study of the structure of attractors of dynamical systems with the objective of estimating their fractal dimension. First we study the case of exponential global attractors of some generalized gradient-like semigroups in a general Banach space, and estimate their fractal dimension in terms of themaximumof the dimension of the local unstablemanifolds of the isolated invariant sets, Lipschitz properties of the semigroup and rate of exponential attraction. We also generalize this result for some special evolution processes, introducing a concept of Morse decomposition with pullback attractivity. Under suitable assumptions, if (A, \'A POT. \') is an attractor-repeller pair for the attractor A of a semigroup {T (t ) : t 0}, then the fractal dimension of A can be estimated in terms of the fractal dimension of the local unstable manifold of \'A POT. \', the fractal dimension of A, the Lipschitz properties of the semigroup and the rate of the exponential attraction. The ingredients of the proof are the notion of generalized gradient-like semigroups and their regular attractors, Morse decomposition and a fine analysis of the structure of the attractors. Also, making use of the skew product semiflow and its Morse decomposition, we give some estimates of the fractal dimension of the pullback attractors of non-autonomous dynamical systemsEste trabalho é dedicado ao estudo da estrutura dos atratores de sistemas dinâmicos com o objetivo de obter estimativas de suas dimensões fractais. Primeiramente estudamos o caso de atratores globais exponenciais de alguns semigrupos gradient-like generalizados em um espaço de Banach geral, e estimamos suas dimensões fractais em termos da máxima dimensão das variedades instáveis locais dos conjuntos invariantes isolados, a propriedades de Lipschitz do semigrupo e da taxa de atração exponencial. Também generalizamos este resultado para alguns processos de evoluções especiais, introduzindo um conceito de decomposição de Morse com atração pullback. Sob hipóteses apropriadas, se (A, \'A POT. \') é um par atrator-repulsor para o atratorA de um semigrupo {T (t ) : t 0}, então a dimensão fractal de A pode ser estimada em termos da dimensão fractal da variedade instável de \'A POT. \', a dimensão fractal de A, as propriedades de Lipschitz do semigrupo e a taxa de atração exponencial. Os ingredientes da demonstração são a noção de semigrupos gradient-like e seus atratores regulares, decomposição de Morse e uma análise fina da estrutura dos atratores. Além disto, fazendo uso do skew product semiflow e sua decomposição de Morse, damos estimativas da dimensão fractal dos atratores pullback de sistêmas dinâmicos não-autônomo

Discrete dynamical systems attractors: fractal dimension and continuity of the structure under perturbations

Neste trabalho, estudamos uma generalização dos semigrupos gradientes, os semigrupos gradiente-like, algumas de suas propriedades e a sua invariância por pequenas perturbações; isto é, pequenas perturbações de sistemas gradiente-like continuam sendo gradiente-like. Como consequência da caracterização dos atratores para este tipo de sistema, estudamos a atração exponencial de atratores. Por fim, estudamos o concetio de dimensão de Hausdorff e dimensão fractal de atratores e apresentamos alguns resultados sobre este assunto, e estudamos a construção de uma nova classe de atratores, os atratores exponenciais fractaisIn this work, we study a generalization of gradient discrete semigroups, the gradientlike semigroups, some of its properties and its invariance under small perturbations; that is, small perturbations of gradient-like semigroups are still gradient-like semigroups. As a consequence of the characterization of the attractors for this sort of semigroups, we study the exponential attraction of attractors. Finally, we study some concepts of Hausdorff dimension and fractal dimension and present some results about this subject, and we studied the construction of a new class of attractors, the exponential fractal attractor