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

Semilinear fractional differential equations: global solutions, critical nonlinearities and comparison results

In this work we study several questions concerning to abstract fractional Cauchy problems of order α ∈ (0, 1). Concretely, we analyze the existence of local mild solutions for the problem, and its possible continuation to a maximal interval of existence. The case of critical nonlinearities and corresponding regular mild solutions is also studied. Finally, by establishing some general comparison results, we apply them to conclude the global well-posedness of a fractional partial differential equation coming from heat conduction theory.Conselho Nacional de Desenvolvimento Científico e TecnológicoCoordenação de aperfeiçoamento de pessoal de nivel superiorFundação de Amparo à Pesquisa do Estado de São PauloMinisterio de EducaciónMinisterio de Ciencia e InnovaciónJunta de Andalucí

Robustness of dynamically gradient multivalued dynamical systems

In this paper we study the robustness of dynamically gradient multivalued semiflows. As an application, we describe the dynamical properties of a family of Chafee-Infante problems approximating a differential inclusion studied in J. M. Arrieta, A. Rodríguez-Bernal and J. Valero, Dynamics of a reaction-diffusion equation with a discontinuous nonlinearity, International Journal of Bifurcation and Chaos, 16 (2006), 2965-2984, proving that the weak solutions of these problems generate a dynamically gradient multivalued semiflow with respect to suitable Morse sets.Ministerio de Educación, Cultura y DeporteMinisterio de Economía y CompetitividadJunta de AndalucíaFundação de Amparo à Pesquisa do Estado de São PauloConselho Nacional de Desenvolvimento Científico e Tecnológic

Gradient-like nonlinear semigroups with infinitely many equilibria and applications to cascade systems

We consider an autonomous dynamical system coming from a coupled system in cascade where the uncoupled part of the system satisfies that the solutions comes from −∞ and goes to ∞ to equilibrium points, and where the coupled part generates asymptotically a gradient-like nonlinear semigroup. Then, the complete model is proved to be also gradient-like. The interest of this extension comes, for instance, in models where a continuum of equilibrium points holds, and for example a Lojasiewicz-Simon condition is satisfied. Indeed, we illustrate the usefulness of the theory with several examples.Fundação de Amparo à Pesquisa do Estado de São PauloConselho Nacional de Desenvolvimento Científico e TecnológicoCoordenação de aperfeiçoamento de pessoal de nivel superiorMinisterio de Ciencia e InnovaciónJunta de AndalucíaMinisterio de Educació

Atractores en dominios tipo dumbbell

En este trabajo, analizamos el comportamiento de la dinámica asintótica de una ecuación de reacción-difusión con condiciones de contorno Neumann homogéneas cuando el dominio se perturba de una forma singular, como en el caso de los dominios de tipo “dumbbell”. Proporcionaremos un marco funcional adecuado para tratar este problema y probaremos que los atractores son semicontinuos superiormente

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

A Non-Autonomous Strongly Damped Wave Equation: Existence and Continuity of the Pullback Attractor

In this paper we consider the strongly damped wave equation with time dependent terms utt − u − γ(t) ut + β"(t)ut = f(u), in a bounded domain ⊂ Rn, under some restrictions on β"(t), γ(t) and growth restrictions on the non-linear term f. The function β"(t) depends on a parameter ε, β"(t) "!0 −→ 0. We will prove, under suitable assumptions, local and global well posedness (using the uniform sectorial operators theory), the existence and regularity of pullback attractors {A"(t) : t ∈ R}, uniform bounds for these pullback attractors, characterization of these pullback attractors and their upper and lower semicontinuity at ǫ = 0

Existence of Pullback Attractors for Pullback Asymptotically Compact Processes

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 nonautonomous evolution processes. An application is considered to damped wave problem with non-autonomous damping