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The monograph is devoted to the study of initial-boundary-value problems for multi-dimensional Sobolev-type equations over bounded domains. The authors consider both specific initial-boundary-value problems and abstract Cauchy problems for first-order (in the time variable) differential equations with nonlinear operator coefficients with respect to spatial variables. The main aim of the monograph is to obtain sufficient conditions for global (in time) solvability, to obtain sufficient conditions for blow-up of solutions at finite time, and to derive upper and lower estimates for the blow-up time. The monograph contains a vast list of references (440 items) and gives an overall view of the contemporary state-of-the-art of the mathematical modeling of various important problems arising in physics. Since the list of references contains many papers which have been published previously only in Russian research journals, it may also serve as a guide to the Russian literature.
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De Gruyter Series in Nonlinear Analysis and Applications 15 Editor in Chief Jürgen Appell, Würzburg, Germany Editors Catherine Bandle, Basel, Switzerland Alain Bensoussan, Richardson, Texas, USA Avner Friedman, Columbus, Ohio, USA Karl-Heinz Hoffmann, Munich, Germany Mikio Kato, Kitakyushu, Japan Umberto Mosco, Worchester, Massachusetts, USA Louis Nirenberg, New York, USA Katrin Wendland, Augsburg, Germany Alfonso Vignoli, Rome, Italy
Alexander B. Al’shin Maxim O. Korpusov Alexey G. Sveshnikov
Blow-up in Nonlinear Sobolev Type Equations
De Gruyter
Mathematics Subject Classification 2010: 35B44, 35D30, 35D35, 35M99, 35Q92, 35Q60, 35J92, 35G20, 35G60.
ISBN 978-3-11-025527-0 e-ISBN 978-3-11-025529-4 ISSN 0941-813X Library of Congress Cataloging-in-Publication Data Al’shin, A. B. Blow-up in nonlinear Sobolev type equations / by A. B. Al’shin, M. O. Korpusov, A. G. Sveshnikov. p. cm. ⫺ (De Gruyter series in nonliniear analysis and applications ; 15) Includes bibliographical references and index. ISBN 978-3-11-025527-0 (alk. paper) 1. Initial value problems ⫺ Numerical solutions. 2. Nonlinear difference equations. 3. Mathematical physics. I. Korpusov, M. O. II. Sveshnikov, A. G. (Aleksei Georgievich), 1924⫺ III. Title. QA378.A47 2011 5151.782⫺dc22 2011003941
Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de. ” 2011 Walter de Gruyter GmbH & Co. KG, Berlin/New York Typesetting: Da-TeX Gerd Blumenstein, Leipzig, www.da-tex.de Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen ⬁ Printed on acid-free paper Printed in Germany www.degruyter.com
Preface
This monograph is devoted to the study of general problems on the global-in-time solvability and blow-up for a finite time of initial-value and initial-boundary-value problems for nonlinear equations of Sobolev type. Our studies together with an outstanding Russian mathematician S. A. Gabov, who prematurely died in 1989, stimulated further work in this direction. Our study of the blow-up of solutions to pseudoparabolic nonlinear equations was considerably stimulated by the classical work of A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhaylov Blow-Up in Quasilinear Parabolic Equations, which influenced the choice of many themes of our monograph. We express a deep appreciation to V. P. Maslov, S. I. Pokhozhaev and N. N. Kalitkin for useful discussion of certain results presented in the monograph. We thank all participants of the scientific seminar “Nonlinear differential equations” and its supervisor Prof. I. A. Shishmaryov for their valuable comments on various sections of the book. The research being the subject of this monogra