Evolution PDEs with Nonstandard Growth Conditions Existence, Uniquen
This monograph offers the reader a treatment of the theory of evolution PDEs with nonstandard growth conditions. This class includes parabolic and hyperbolic equations with variable or anisotropic nonlinear structure. We develop methods for the study of s
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Stanislav Antontsev Sergey Shmarev
Evolution PDEs with Nonstandard Growth Conditions Existence, Uniqueness, Localization, Blow-up
Atlantis Studies in Differential Equations Volume 4
Series editor Michel Chipot, Zürich, Switzerland
Aims and Scope of the Series The “Atlantis Studies in Differential Equations” publishes monographs in the area of differential equations, written by leading experts in the field and useful for both students and researchers. Books with a theoretical nature will be published alongside books emphasizing applications. For more information on this series and our other book series, please visit our website at: www.atlantis-press.com/publications/books AMSTERDAM – PARIS – BEIJING ATLANTIS PRESS Atlantis Press 29, avenue Laumière 75019 Paris, France
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Stanislav Antontsev Sergey Shmarev •
Evolution PDEs with Nonstandard Growth Conditions Existence, Uniqueness, Localization, Blow-up
Stanislav Antontsev Center for Mathematics and Fundamental Applications University of Lisbon Lisbon Portugal
Sergey Shmarev Department of Mathematics University of Oviedo Oviedo, Asturias Spain
ISSN 2214-6253 ISSN 2214-6261 (electronic) Atlantis Studies in Differential Equations ISBN 978-94-6239-111-6 ISBN 978-94-6239-112-3 (eBook) DOI 10.2991/978-94-6239-112-3 Library of Congress Control Number: 2015935210 © Atlantis Press and the author(s) 2015 This book, or any parts thereof, may not be reproduced for commercial purposes in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system known or to be invented, without prior permission from the Publisher. Printed on acid-free paper
This work is dedicated to our families, Tamara, Nikolay, Stanislav and Elena, Dmitry, Andrey, to whom we owe so much.
Preface
This monograph is a contribution to the theory of second order quasilinear parabolic and hyperbolic equations with the nonlinear structure that may change from one point to another in the problem domain. In the past decade, there was an impetuous growth of interest in the study of such equations, which appear in a natural way in the mathematical modeling of various real-world phenomena and give rise to challenging mathematical problems. The aim of this work is to give an account of the known results on existence, uniqueness, and qualitative properties of solutions. The parabolic equations studied below can be conventionally divided into several groups. Chaps. 2 and 3 are devoted to study the generalized porous medium equation ut ¼ div jujmðx;tÞ ru þ f ðx; tÞ
ð1Þ
with a given exponent mðx; tÞ [ 1 and its generalizations, such as equations with lower order terms or anisotropic equations. We establish conditions of existence and uniqueness of weak solutions and show that for definite ranges of the exponent mðx; tÞ the solutions exhibit properties typical for the solutions of equations with constant m, those of the finite speed of propagation and extinction in finite tim
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