Asymptotical dynamics of evolution processes

Abstract

The goal of this project is the study of the asymptotical dynamics of evolution processes $\{T(t,s): t\geqslant s\}$ in infinite dimensional Banach spaces. More specifically, we are interested in two fundamental aspects to the study of asymptotical dynamics; namely, the study of internal structures of the pullback attractors and the study of properties of theses structures, which can give us information about the pullback attractor. In this point, for example, appear the continuity of pullback attractors through the continuity of local unstable sets in gradient-like processes, as well as the computation of the fractal dimension of pullback attractors in terms of the fractal dimension of the local unstable sets. We intend to explore an approximation process of semilinear problems (parabolic and hyperbolic PDE's) where the unbounded operators are replaced by their Yosida approximations in order to study the transport of dynamical informations from the perturbed problem (with the Yosida approximation) to the original limit problem. These approximations appear in the literature in several examples (although the limit passage is not, in general, explored to obtain informations about the limit problem). Some examples are the Navier-Stokes-Voight equation and the strongly damped wave equation. Here, our goal is to obtain informations about the ``continuity'' of the dynamic structures under these perturbations.In reference to the study of the dynamical structures of evolution processes, one of the obstacles found is strongly related to the characterization of the exponential dichotomy (which replaces de concept of hyperbolicity for non-autonomous processes) and its stability under perturbations (asynchronous or with phase), since the asymptotic behavior in the forward dynamics is related to a family of homogenized processes. We approach these questions in examples (infinite dimensional) with the objective to create a general theory, as the one found in Coppel (Lecture 06).Giving continuity to the developed project during the PhD., we will perform a deeper study of the dynamical properties of the pullback attractors to non-autonomous dynamical systems via skew product semiflow, transporting informations from the driving system to the skew product semiflow and to the associated evolution process.