Nonlinear Partial Differential Equations Asymptotic Behavior of Solu
The main focus of this textbook, in two parts, is on showing how self-similar solutions are useful in studying the behavior of solutions of nonlinear partial differential equations, especially those of parabolic type. The exposition moves systematically f
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Mi-Ho Giga
•
Yoshikazu Giga
•
J¨urgen Saal
Nonlinear Partial Differential Equations Asymptotic Behavior of Solutions and Self-Similar Solutions
Birkh¨auser Boston • Basel • Berlin
Mi-Ho Giga Graduate School of Mathematical Sciences University of Tokyo Komaba 3-8-1, Meguro-ku Tokyo 153-8914 Japan [email protected]
Yoshikazu Giga Graduate School of Mathematical Sciences University of Tokyo Komaba 3-8-1, Meguro-ku Tokyo 153-8914 Japan [email protected]
J¨urgen Saal Technische Universit¨at Darmstadt Center of Smart Interfaces Petersenstraße 32 64287 Darmstadt Germany [email protected]
ISBN 978-0-8176-4173-3 e-ISBN 978-0-8176-4651-6 DOI 10.1007/978-0-8176-4651-6 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010928841 Mathematics Subject Classification (2010): Primary: 35B40, 35C06. Secondary: 35Q30, 76D05, 35K05, 35G55, 26D10, 42B20 c Springer Science+Business Media, LLC 2010
All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Birkh¨auser is part of Springer Science+Business Media (www.birkhauser.com)
In memory of Professor Tetsuro Miyakawa – with our profound admiration
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Part I Asymptotic Behavior of Solutions of Partial Differential Equations 1
Behavior Near Time Infinity of Solutions of the Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Asymptotic Behavior of Solutions Near Time Infinity . . . . . . . . 1.1.1 Decay Estimate of Solutions . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Lp -Lq Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Derivative Lp -Lq Estimates . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Theorem on Asymptotic Behavior Near Time Infinity . . 1.1.5 Proof Using Representation Formula of Solutions . . . . . . 1.1.6 Integral Form of the Mean Value Theorem . . . . . . . . . . . . 1.2 Structure of Equations and Self-Similar Solutions . . . . . . . . . . . . 1.2.1 Invariance Under Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Conserved Quantity for the Heat Equation . . . . . . . . .
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