Blow-up Theories for Semilinear Parabolic Equations
There is an enormous amount of work in the literature about the blow-up behavior of evolution equations. It is our intention to introduce the theory by emphasizing the methods while seeking to avoid massive technical computations. To reach this goal, we u
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2018
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Bei Hu 胡钡
Blow-up Theories for Semilinear Parabolic Equations
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Bei Hu University of Notre Dame Department of Applied and Computational Mathematics and Statistics 46556 Notre Dame Indiana USA [email protected]
ISBN 978-3-642-18459-8 e-ISBN 978-3-642-18460-4 DOI 10.1007/978-3-642-18460-4 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2011924396 Mathematics Subject Classification (2011): 35K10, 35K15, 35K57, 35K58, 35K51, 35J61, 35J15, 35J65 c Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
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Yi Cheng 程怡
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Preface
I am grateful to Professor Zhengce Zhang (张政策) and his students from Xi’an Jiaotong university for correcting many typographic errors in my 2005 version of the lecture notes from their graduate reading seminar in the past few years. I am also grateful to the reviewers who offered constructive comments and suggested improvements of the manuscript. I would like to thank the editors of the LNM series for their suggestions, which improved the presentation of the materials and made the lecture notes more friendly to students. I would also like to thank Timothy McCoy for reading through the entire manuscript. Notre Dame, IN Fall 2010
Bei Hu
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Preface
These are the lecture notes of a course given at Xi’an Jiaotong university in summer of 2005. They are intended for beginning graduate students who have finished a first-year graduate course in basic partial differential equations. The prerequisites include an understanding of the basic theory of the secondorder equations such as 1. Maximum principles, basic existence and uniqueness theorems. 2. A priori estimates such as the Schauder estimates, the Lp estimates, the De Giorgi–Nash–Moser estimates, the Krylov–Safanov estimates. 3. The fixed point theorems. There is an enormous amount of work in the literature about the blow-up behavior of evolution equations. It is our intention to introduce the theory by emphasizing the methods while avoiding the massiv
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