05056nam a22005655i 4500 978-1-4614-4581-4 DE-He213 20210706074133.0 cr nn 008mamaa 120928s2013 xxu| s |||| 0|eng d 9781461445814 978-1-4614-4581-4 10.1007/978-1-4614-4581-4 doi QA370-380 PBKJ bicssc MAT007000 bisacsh PBKJ thema 515.353 23 Carvalho, Alexandre. author. aut http://id.loc.gov/vocabulary/relators/aut Attractors for infinite-dimensional non-autonomous dynamical systems [electronic resource] / by Alexandre Carvalho, José A. Langa, James Robinson. 1st ed. 2013. New York, NY : Springer New York : Imprint: Springer, 2013. XXXVI, 412 p. online resource. text txt rdacontent computer c rdamedia online resource cr rdacarrier text file PDF rda Applied Mathematical Sciences, 0066-5452 ; 182 The pullback attractor -- Existence results for pullback attractors -- Continuity of attractors -- Finite-dimensional attractors -- Gradient semigroups and their dynamical properties -- Semilinear Differential Equations -- Exponential dichotomies -- Hyperbolic solutions and their stable and unstable manifolds -- A non-autonomous competitive Lotka-Volterra system -- Delay differential equations.-The Navier–Stokes equations with non-autonomous forcing.-  Applications to parabolic problems -- A non-autonomous Chafee–Infante equation -- Perturbation of diffusion and continuity of attractors with rate -- A non-autonomous damped wave equation -- References -- Index.-. This book treats the theory of pullback attractors for non-autonomous dynamical systems. While the emphasis is on infinite-dimensional systems, the results are also applied to a variety of finite-dimensional examples.   The purpose of the book is to provide a summary of the current theory, starting with basic definitions and proceeding all the way to state-of-the-art results. As such it is intended as a primer for graduate students, and a reference for more established researchers in the field.   The basic topics are existence results for pullback attractors, their continuity under perturbation, techniques for showing that their fibres are finite-dimensional, and structural results for pullback attractors for small non-autonomous perturbations of gradient systems (those with a Lyapunov function).  The structural results stem from a dynamical characterisation of autonomous gradient systems, which shows in particular that such systems are stable under perturbation. Application of the structural results relies on the continuity of unstable manifolds under perturbation, which in turn is based on the robustness of exponential dichotomies: a self-contained development of  these topics is given in full. After providing all the necessary theory the book treats a number of model problems in detail, demonstrating the wide applicability of the definitions and techniques introduced: these include a simple Lotka-Volterra ordinary differential equation, delay differential equations, the two-dimensional Navier-Stokes equations, general reaction-diffusion problems, a non-autonomous version of the Chafee-Infante problem, a comparison of attractors in problems with perturbations to the diffusion term, and a non-autonomous damped wave equation. Alexandre N. Carvalho is a Professor at the University of Sao Paulo, Brazil. José A. Langa is a Profesor Titular at the University of Seville, Spain. James C. Robinson is a Professor at the University of Warwick, UK. Partial differential equations. Dynamics. Ergodic theory. Manifolds (Mathematics). Complex manifolds. Partial Differential Equations. https://scigraph.springernature.com/ontologies/product-market-codes/M12155 Dynamical Systems and Ergodic Theory. https://scigraph.springernature.com/ontologies/product-market-codes/M1204X Manifolds and Cell Complexes (incl. Diff.Topology). https://scigraph.springernature.com/ontologies/product-market-codes/M28027 Langa, José A. author. aut http://id.loc.gov/vocabulary/relators/aut Robinson, James. author. aut http://id.loc.gov/vocabulary/relators/aut SpringerLink (Online service) Springer Nature eBook Printed edition: 9781461445807 Printed edition: 9781461445821 Printed edition: 9781489991768 Applied Mathematical Sciences, 0066-5452 ; 182 https://doi.org/10.1007/978-1-4614-4581-4 ZDB-2-SMA ZDB-2-SXMS Mathematics and Statistics (SpringerNature-11649) Mathematics and Statistics (R0) (SpringerNature-43713)