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Dynamical systems and their attractors under perturbations

Abstract

The dynamical systems that we seek to understand are those derived from differential equations in Banach spaces, which includes ordinary differential equations and partial semilinear (or quasilinear) evolutionary differential equations. The treatment that we give to these models has its origin in the spectral theory, via operational calculus, semigroup theory of linear operators and the variation of constants formula. In this way, the semilinear (and quasilinear) evolutionary partial differential equations that we consider are ordinary differential equations in Banach spaces. In general, these equations are classified into two groups, that is, as parabolic differential equations, when the associated linear part generates a strongly continuous and analytical semigroup of linear operators (Navier-Stokes, Heat, Fitshugh-Nagumo, Cahn-Hilliard, etc.), and as hyperbolic differential equations, when a linear part generates only a strongly continuous semigroup of linear operators (Retarded, Wave, Schödinger, etc.). For the studied equations (which also include couplings of the main types), we also consider the effect of impulses (impulsive equations) or noise (random/stochastic equations). In several of these models, the study of linear and semilinear elliptic problems plays a fundamental role, particularly for the study of parabolic and hyperboilic differential equations. On one hand, because the linear elliptic operators make up (is a part or all) the generator of the associated semigroup of linear operators, and on the other hand, because the solutions to the semilinear elliptic problem make up (are a part or all) the stationary or equilibrium solutions. Throughout many years we have helped to build a general theory to understand how these dynamical systems behave under perturbations. Our previous and proposed contributions range form local well posedness to structural stability of global attractors under regular and singular perturbations, autonomous or not. (AU)

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