Global Attractors Of Non-autonomous Dissipative Dynamical Systems

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. GLOBAL ATTRACTORS OF NON-AUTONOMOUS DISSIPATIVE DYNAMICAL SYSTEMS Copyright © 2004 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 981-256-028-9 Printed in Singapore. Dedicated to my wife Ivanna and my children Olga and Anatoli vi Preface In the qualitative theory of differential equations non-local problems play an important role, especially in regard to questions of boundedness, periodicity, almost periodicity, Poisson stability, asymptotic behaviour, dissipativity, etc. The present work takes a similar approach and is dedicated to the study of abstract non-autonomous dissipative dynamical systems and their application to differential equations. In applications there often occur systems u0 = f (t, u), (0.1) which have every one of their solutions driven into fixed bounded domain and kept there under further increase of time, becouse of natural dissipation. Such systems are called dissipative ones [135]-[137],[270]-[272],[325], [326]. Solutions of dissipative systems are called limit (finally) bounded [325, 326]. Dynamical systems occur in hydrodynamics studying turbulent phenomena, meteorology, oceanography, theory of oscillations, biology, radio engineering and other domains of sciences and engineering technics related to the study of asymptotic behaviour. Lately interest in dissipative systems increased even more because of intensive elaboration of strange attractors (see, e.g.,[143, 239, 296, 306]). The study of the dissipative systems there are dedicated plenty of works, beginning from the classical works of N. Levinson. Among works on dissipative systems of ordinary differential equations two directions can be made out. To the first belong works which contain some conditions assuring the dissipativity of system (0.1), some class or concrete system, representing theoretical or applied interest. Examples are works of P. V. Atrashenok [9], B. P. Demidovich [135]-[137], V. I. Zubov [336]-[338], V. M. Matrosov [251], V. V. Nemytski [259]-[260], V. A. Pliss [270], V. N. Schennikov [298, 299], C. Corduneanu [125], N. Levinson [237], N. Pavel [264]-[266], R. Reissig [272], T. Talpalaru [309] and a lot of other authors. To the second direction belong works in which are studied inner conditions vii viii Global Attractors of Non-autonomous Dissipative Dynamical Systems of dissipative systems, that is, conditions relating to the character of behaviour of solution of the system when assuming its dissipativity, for different classes of differential equations. Among them are the works of V. M. Gershtein [159]-[163], V. V. Zhikov [331]-[332], I. L. Zinchenko [334], M. A. Krasnoselsky [159], S. Yu. Pilyugin [269], V. A. Pliss [270]-[271], M. L. Cartwright and J. E. Littlewood [44][46], N. Levinson [237], G. Fusco and M. Oliva [157], J. Skowronski and S. Ziemba [307] and other authors. For a compact map on a Banach space, V. M. Gerstein [162], V. M. Gerstein and M. A. Krasnoselskii [159] investigated t