Differential Equations Inverse and Direct Problems Copyright © 2006 Taylor & Francis Group, LLC PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes EXECUTIVE EDITORS Earl J Taft Rutgers University Piscataway, New Jersey Zuhair Nashed University of Central Florida Orlando, Florida EDITORIAL BOARD M S Baouendi University of California, San Diego Jane Cronin Rutgers University Jack K Hale Georgia Institute of Technology S Kobayashi University of California, Berkeley Marvin Marcus University of California, Santa Barbara W S Massey Yale University Anil Nerode Cornell University Copyright © 2006 Taylor & Francis Group, LLC Freddy van Oystaeyen University of Antwerp, Belgium Donald Passman University of Wisconsin, Madison Fred S Roberts Rutgers University David L Russell Virginia Polytechnic Institute and State University Walter Schempp Universität Siegen Mark Teply University of Wisconsin, Milwaukee LECTURE NOTES IN PURE AND APPLIED MATHEMATICS Recent Titles G 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Goeters and O Jenda, Abelian Groups, Rings, Modules, and Homological Algebra J Cannon and B Shivamoggi, Mathematical and Physical Theory of Turbulence A Favini and A Lorenzi, Differential Equations: Inverse and Direct Problems Copyright © 2006 Taylor & Francis Group, LLC Differential Equations Inverse and Direct Problems Edited by Angelo Favini Università degli Studi di Bologna Italy Alfredo Lorenzi Università degli Studi di Milano Italy Boca Raton London New York Chapman & Hall/CRC is an imprint of the Taylor & Francis Group, an informa business Copyright © 2006 Taylor & Francis Group, LLC C6048_Discl.fm Page Thursday, March 16, 2006 11:29 AM Published in 2006 by Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group No claim to original U.S Government works Printed in the United States of America on acid-free paper 10 International Standard Book Number-10: 1-58488-604-8 (Softcover) International Standard Book Number-13: 978-1-58488-604-4 (Softcover) Library of Congress Card Number 2006008692 This book contains information obtained from authentic and highly regarded sources Reprinted material is quoted with permission, and sources are indicated A wide variety of references are listed Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Library of Congress Cataloging-in-Publication Data Favini, A (Angelo), 1946Differential equations : inverse and direct problems / Angelo Favini, Alfredo Lorenzi p cm (Lecture notes in pure and applied mathematics ; v 251) Includes bibliographical references ISBN 1-58488-604-8 Differential equations Inverse problems (Differential equations) Banach spaces I Lorenzi, Alfredo II Title III Series QA371.F28 2006 515'.35 dc22 2006008692 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com Taylor & Francis Group is the Academic Division of Informa plc Copyright © 2006 Taylor & Francis Group, LLC and the CRC Press Web site at http://www.crcpress.com Preface The meeting on Differential Equations: Inverse and Direct Problems was held in Cortona, June 21-25, 2004 The topics discussed by well-known specialists in the various disciplinary fields during the Meeting included, among others: differential and integrodifferential equations in Banach spaces, linear and nonlinear theory of semigroups, direct and inverse problems for regular and singular elliptic and parabolic differential and/or integrodifferential equations, blow up of solutions, elliptic equations with Wentzell boundary conditions, models in superconductivity, phase transition models, theory of attractors, GinzburgLandau and Schrăodinger equations and, more generally, applications to partial differential and integrodifferential equations from Mathematical Physics The reports by the lecturers highlighted very recent, interesting and original research results in the quoted fields contributing to make the Meeting very attractive and stimulating also to younger participants After a lot of discussions related to the reports, some of the senior lecturers were asked by the organizers to provide a paper on their contribution or some developments of them The present volume is the result of all this In this connection we want to emphasize that almost all the contributions are original and are not expositive papers of results published elsewhere Moreover, a few of the contributions started from the discussions in Cortona and were completed in the very end of 2005 So, we can say that the main purpose of the editors of this volume has consisted in stimulating the preparation of new research results As a consequence, the editors want to thank in a particular way the authors that have accepted this suggestion Of course, we warmly thank the Italian Istituto Nazionale di Alta Matematica that made the Meeting in Cortona possible and also the Universit´a degli Studi di Milano for additional support Finally, the editors thank the staff of Taylor & Francis for their help and useful suggestions they supplied during the preparation of this volume Angelo Favini and Alfredo Lorenzi Bologna and Milan, December 2005 vii Copyright © 2006 Taylor & Francis Group, LLC Contents M Al-Horani and A Favini: Degenerate first order identification problems in Banach spaces V Berti and M Fabrizio: A nonisothermal dynamical Ginzburg-Landau model of superconductivity Existence and uniqueness theorems 17 F Colombo, D Guidetti and V Vespri: Some global in time results for integrodifferential parabolic inverse problems 35 A Favini, G Ruiz Goldstein, J A Goldstein, and S Romanelli: Fourth order ordinary differential operators with general Wentzell boundary conditions 59 A Favini, R Labbas, S Maingot, H Tanabe and A Yagi: Study of elliptic differential equations in UMD spaces 73 A Favini, A Lorenzi and H Tanabe: Degenerate integrodifferential equations of parabolic type 91 A Favini, A Lorenzi and A Yagi: Exponential attractors for semiconductor equations 111 S Gatti and M Grasselli: Convergence to stationary states of solutions to the semilinear equation of viscoelasticity 131 S Gatti and A Miranville: Asymptotic behavior of a phase field system with dynamic boundary conditions 149 M Geissert, B Grec, M Hieber and E Radkevich: The model-problem associated to the Stefan problem with surface tension: an approach via Fourier-Laplace multipliers 171 G Ruiz Goldstein, J A Goldstein and I Kombe: The power potential and nonexistence of positive solutions 183 A Lorenzi and H Tanabe: Inverse and direct problems for nonautonomous degenerate integrodifferential equations of parabolic type with Dirichlet boundary conditions 197 ix Copyright © 2006 Taylor & Francis Group, LLC x F Luterotti, G Schimperna and U Stefanelli: Existence results for a phase transition model based on microscopic movements 245 N Okazawa: Smoothing effects and strong L2 -wellposedness in the complex Ginzburg-Landau equation 265 Copyright © 2006 Taylor & Francis Group, LLC Contributors Mohammed Al-Horani Department of Mathematics, University of Jordan, Amman, Jordan horani@ju.edu.jo Valeria Berti Department of Mathematics, University of Bologna, Piazza di Porta S.Donato 5, 40126 Bologna, Italy berti@dm.unibo.it Fabrizio Colombo Department of Mathematics, Polytechnic of Milan, Via Bonardi 9, 20133 Milan, Italy fabcol@mate.polimi.it Mauro Fabrizio Department of Mathematics, University of Bologna, Piazza di Porta S.Donato 5, 40126 Bologna, Italy fabrizio@dm.unibo.it Angelo Favini Department of Mathematics, University of Bologna, Piazza di Porta S Donato 5, 40126 Bologna, Italy favini@dm.unibo.it Stefania Gatti Department of Mathematics, University of Ferrara, Via Machiavelli 35, Ferrara, Italy stefania.gatti@.unife.it Matthias Geissert Department of Mathematics, Technische Universităat Darmstadt, Darmstadt, Germany geissert@mathematik.tu-darmstadt.de Gisle Ruiz Goldstein Department of Mathematical Sciences University of Memphis, Memphis Tennessee 38152 ggoldste@memphis.edu Jerome A Goldstein Department of Mathematical Sciences University of Memphis, Memphis Tennessee 38152 jgoldste@memphis.edu Maurizio Grasselli Department of Mathematics, Polytechnic of Milan Via Bonardi 9, 20133 Milan, Italy maugra@mate.polimi.it B´ er´ enice Grec Department of Mathematics, Technische Universităat Darmstadt, Darmstadt, Germany berenice.grec@web.de xi Copyright © 2006 Taylor & Francis Group, LLC xii Davide Guidetti Department of Mathematics, University of Bologna, Piazza di Porta S Donato 5, 40126 Bologna, Italy guidetti@dm.unibo.it Matthias Hieber Department of Mathematics, Technische Universităat Darmstadt, Darmstadt, Germany hieber@mathematik.tu-darmstadt.de Ismail Kombe Mathematics Department, Oklahoma City University 2501 North Blackwelder, Oklahoma City OK 73106-1493, U.S.A ikombe@okcu.edu Rabah Labbas Laboratoire de Math´ematiques, Facult´e des Sciences et Techniques, Universit´e du Havre, B.P 540, 76058 Le Havre Cedex, France rabah.labbas@univ-lehavre.fr Alfredo Lorenzi Department of Mathematics, Universit`a degli Studi di Milano, via C Saldini 50, 20133 Milano, Italy lorenzi@mat.unimi.it Fabio Luterotti Department of Mathematics, University of Brescia Via Branze 38, 25123 Brescia, Italy luterott@ing.unibs.it St´ ephane Maingot Laboratoire de Math´ematiques, Facult´e des Sciences et Techniques, Universit´e du Havre, B.P 540, 76058 Le Havre Cedex, France rabah.labbas@univ-lehavre.fr Alain Miranville Laboratoire de Math´ematiques et Applications, UMR CNRS 6086 - SP2MI, Boulevard Marie et Pierre Curie - Tlport F-86962 Chasseneuil Futuroscope Cedex, France miranv@math.univ-poitiers.fr Noboru Okazawa Department of Mathematics, Science University of Tokyo Wakamiya-cho 26, Shinjuku-ku Tokyo 162-8601, Japan okazawa@ma.kagu.sut.ac.jp Evgeniy Radkevich Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia evrad@land.ru Silvia Romanelli Department of Mathematics, University of Bari Via E Orabona 4, 70125 Bari, Italy romans@dm.uniba.it Giulio Schimperna Department of Mathematics, University of Pavia Via Ferrata 1, 27100 Pavia, Italy giulio@dimat.unipv.it Ulisse Stefanelli IMATI, Universit`a degli Studi di Pavia Via Ferrata 1, 27100 Pavia, Italy ulisse@imati.cnr.it Copyright © 2006 Taylor & Francis Group, LLC 274 N Okazawa As for the maximality of (κ + iβ)B (i.e., R(1 + (κ + iβ)B) = X, which is equivalent to that of B) it suffices to note that the m-accretivity of B is reduced to that of the mapping z → |z|q−2 z on C (cf [28, Lemma 3.1]) Thus, as mentioned in Section 2, for every v ∈ X there is a unique solution uε ∈ D(B) to (AStP)ε Next, it follows from (3.4) that λ Sε u X ≤ (λ + iα)Sε u + (κ + iβ)Bu X, ∀ u ∈ D(B), λ > 0, while the boundedness of {uε } follows from the condition D(S) ∩ D(B) = ∅ In this way (AStP)ε yields the boundedness of {Sε uε }: λ Sε uε X ≤ v X + uε X , so that we can show that {uε } satisfies Cauchy condition for convergence Since both of {Sε uε } and {Buε } converge weakly as ε ↓ 0, u := limε↓0 uε is proved to be a unique solution to (AStP) (the weak closedness of S and the demi-closedness of B in X) This concludes the m-accretivity of A + γ = (λ + iα)S + (κ + iβ)B and the generation of the corresponding semigroup {U (t)} on D(A) However, Theorem 3.1 guarantees the existence of unique strong solutions to (ACP) just for initial values from D(A) = D(S) ∩ D(B) (usual theory of nonlinear (quasi-)contraction semigroups) Now, under the setting of Theorem 3.2, we consider the sequence {un (·)} = {U (·)u0,n } of solutions to (ACP) with initial value {u0,n } ⊂ D(A), where {u0,n } is an approximate sequence for u0 ∈ D(A):   Dt un + (λ + iα)(Sun )(t) + (κ + iβ)∂ψ(un (t)) − γun (t) = (ACP)n a.e on R+ ,   un (0) = u0,n We see from (3.5) that U (·)u0 = lim un (·) in C([0, T ]; X) for every T > n→∞ The proof of the smoothing effect U (t)u0 = lim un (t) ∈ D(A) (t > 0) is n→∞ based on the equality: Dt (ψ ◦ un )(t) = Re ∂ψ(un (t)), Dt un (t) X , for a.e on (0, ∞) Setting η := (λ2 + α2 )−1 λκ, we can show that {Sun }, {∂ψ(un )} are bounded in L2 (T −1 , T ; X) for all T > 1: T T −1 ( (Sun )(t) X + η ∂ψ(un (t)) X ) dt ≤ M T (1 + T )ekT u0,n X, where M > and k > 2γ+ := max{γ, 0} are two constants In view of (ACPeq)n , {Dt un } is also bounded in L2 (T −1 , T ; X) Therefore U (t)u0 , the limit of un (t), satisfies (ACPeq) a.e on (T −1 , T ) In fact, Dt and S are weakly closed, while ∂ψ is demi-closed in L2 (T −1 , T ; X) Since T > is arbitrary, U (t)u0 is a strong solution to (ACP) But, what we have proved is U (t)u0 ∈ D(A) for a.e on (0, ∞), ∀ u0 ∈ D(A) Copyright © 2006 Taylor & Francis Group, LLC (3.7) Strong L2 -wellposedness in the complex Ginzburg-Landau equation 275 Nevertheless, we can remove the term “a.e.” in (3.7) In fact, we can choose t0 > in “U (t0 )u0 ∈ D(A)” as small as possible so that we can take U (t0 )u0 as a new initial value in Theorem 3.1 to conclude that U (t)u0 ∈ D(A) (t ≥ t0 ) In what follows we shall verify the inequalities (3.4) and (3.2) when S, B are given by (2.1), (2.2), respectively Verification of (3.4) Let z, y ∈ C with z = y Then in [22, Lemma 2.2] Liskevich-Perelmuter derived the following remarkable inequality (a simpler proof is given in [29, Lemma 2.1]): |q − 2| |Im (z − y)(|z|q−2 z − |y|q−2 y)| , ≤ √ Re (z − y)(|z|q−2 z − |y|q−2 y) q−1 < q < +∞ Setting z := v(x), y := w(x) and integrating it on Ω, we have a new inequality for complex-valued functions: for v, w ∈ Lq (Ω) (v = w) |Im v − w, |v|q−2 v − |w|q−2 w Re v − w, |v|q−2 v − |w|q−2 w Lq ,Lq Lq ,Lq | |q − 2| , ≤ √ q−1 1 = 1; + q q (3.8) note that |v|q−2 v ∈ Lq (Ω) ∀ v ∈ Lq (Ω) Next, we utilize the m-accretive realization Aq of the minus Laplacian (with Dirichlet condition) in Lq (Ω): Aq := −∆ with D(Aq ) := W 2,q (Ω) ∩ W01,q (Ω), < q < ∞ Then we can show that Aq is sectorial of type S(tan ω), ω := tan−1 |q − 2| √ , q−1 in the sense of Goldstein [14, Definition 1.5.8, p 37] (cf Henry [16, p 32]; see also [26]): for all u ∈ D(Aq ), |Im Aq u, |u|q−2 u Lq ,Lq |q − 2| Re Aq u, |u|q−2 u |≤ √ q−1 Lq ,Lq (3.9) It seems to be a surprise that the constant in (3.8) and (3.9) coincides Such a coincidence will be explained as a proof of (3.4) in the rest of this section Now we want to prove (3.9) with Aq replaced with its Yosida approximation Aq, ε : for all v ∈ Lq (Ω), |Im Aq, ε v, |v|q−2 v Lq ,Lq |q − 2| Re Aq, ε v, |v|q−2 v |≤ √ q−1 Lq ,Lq (3.10) The proof of (3.10) is simply performed by combining (3.8) with (3.9) Noting that (1 + εAq )−1 v ∈ Lq (Ω) for every v ∈ Lq (Ω), we can express the pairing Copyright © 2006 Taylor & Francis Group, LLC 276 N Okazawa Aq, ε v, |v|q−2 v Lq ,Lq Aq, ε v, |v|q−2 v as follows: Lq ,Lq = Aq (1 + εAq )−1 v, |(1 + εAq )−1 v|q−2 (1 + εAq )−1 v Lq ,Lq + ε−1 v − (1 + εAq )−1 v, |v|q−2 v − |(1 + εAq )−1 v|q−2 (1 + εAq )−1 v Lq ,Lq , where we have used the definition of Yosida approximation Here we interpret one and the same element (1 + εAq )−1 v on the right-hand side in two different ways, namely as u := (1 + εAq )−1 v ∈ D(Aq ) in the first term and as w := (1 + εAq )−1 v ∈ Lq (Ω) in the second term In this way we can write down the pairing as follows: Aq, ε v, |v|q−2 v = Aq u, |u|q−2 u Lq ,Lq Lq ,Lq + ε−1 v − w, |v|q−2 v − |w|q−2 w Lq ,Lq Applying (3.9) and (3.8) to the first and second terms on the right-hand side, respectively, we can obtain (3.10) Finally, let u ∈ D(B), q ≥ Then we have (Sε u, Bu)L2 = Aq, ε u, |u|q−2 u Lq ,Lq , ∀ u ∈ D(B) (3.11) In fact, noting that D(B) = L2 (Ω) ∩ L2(q−1) (Ω) ⊂ Lq (Ω), we can conclude that Sε u = Aq, ε u ∈ L2 (Ω) ∩ Lq (Ω) and Bu = |u|q−2 u ∈ L2 (Ω) ∩ Lq (Ω) This proves (3.11) By virtue of (3.10) we can obtain (3.4) with X = L2 (Ω) and q−2 ; tan ωq = cq := √ q−1 in this connection see also Remark 4.2 below Verification of (3.2) (3.2) is also derived from (3.8) in the same way as above Proof of Theorem 1.2 Given a nonnegative selfadjoint operator S in a complex Hilbert space X, let ϕ(v) be the proper lower semi-continuous convex function defined as ϕ(v) := (1/2) S 1/2 v X +∞ Copyright © 2006 Taylor & Francis Group, LLC if v ∈ D(ϕ) := D(S 1/2 ), otherwise, Strong L2 -wellposedness in the complex Ginzburg-Landau equation 277 where S 1/2 is the square root of S = ∂ϕ Then the equation in (ACP)ε is given by (ACPeq)ε Dt uε (t) + (λ + iα)∂ϕ(uε (t)) + (κ + iβ)∂ψε (uε (t)) − γuε (t) = a.e on R+ , where ψ ≥ is another proper lower semi-continuous convex function on X Now we introduce five conditions on ϕ and ψ (or S = ∂ϕ and ∂ψ) (A1) {u ∈ X; ϕ(u) ≤ c} is compact in X ∀ c > (A2) ∃ q ∈ [2, ∞); ψ(ζu) = |ζ|q ψ(u) ∀ u ∈ D(ψ), ∀ ζ ∈ C with Re ζ > (A3) ∃ θ ∈ [0, 1]; ∀ (λ, κ, β) ∈ R+ × R+ × R ∃ c1 = c1 (λ, κ, β) > such that for u ∈ D(S) and ε > 0, |(Su, ∂ψε (u))X | ≤ c1 ψ(Jε u)θ ϕ(u) + λ κ2 + β2 Su X, (4.1) where Jε := (1 + ε∂ψ)−1 (A4) D(S) ⊂ D(∂ψ) and ∃ c2 > such that ∂ψ(u) X ≤ c2 ( Su X + u X) ∀ u ∈ D(S) (A5) ∀ (λ, κ, β) ∈ R+ × R+ × R ∃ c3 = c3 (λ, κ, β) > such that |(∂ψ(u) − ∂ψ(v), u − v)X | ψ(u) + ψ(v) θ u − v 2X + ≤ c3 2λ κ2 + β2 ϕ(u − v) (4.2) for u, v ∈ D(∂ψ) ∩ D(ϕ) Here θ ∈ [0, 1] is the same constant as in (A3) As a consequence of condition (A1) we can use compactness methods to prove the convergence of approximate solutions to (ACP)ε In (CGL) we assume that Ω ⊂ Rn is bounded so that Rellich’s theorem is available Now we can state two abstract theorems which lead us to Theorem 1.2 The first theorem is concerned with the existence of solutions to (ACP) THEOREM 4.1 ([30]) Let S be a nonnegative selfadjoint operator in X, ψ a proper lower semi-continuous convex function on X Assume that conditions (A1)–(A4) are satisfied Then for any initial value u0 ∈ D(ϕ) ∩ D(ψ) there exists a strong solution u to (ACP) such that u ∈ C 0,1/2 ([0, T ]; X) ∀ T > Copyright © 2006 Taylor & Francis Group, LLC 278 N Okazawa In connection with (CGL) this theorem is interpreted as follows To avoid the restriction on the coefficient κ + iβ of the nonlinear term |u|q−2 u the size of the power q − is strictly restricted in Theorem 4.1 (cf (1.4)) To see this θ ∈ [0, 1] in condition (A3) is a key constant In the process to verify (A3) θ is given by θ = fN (q) := 2(q − 2) , 2q − N (q − 2) 2≤q ≤2+ ; N (4.3) note that = fN (2) ≤ fN (q) ≤ fN (2 + 4/N ) = To the contrary, in [29, Theorem 4.1] there is a strong restriction on κ + iβ, whereas the power is not restricted Conditions (A1), (A2) are common in these theorems In the second theorem an additional condition (A5) guarantees the wellposedness of (ACP) THEOREM 4.2 ([30]) Let S and ψ be the same as in Theorem 4.1 Assume that conditions (A1)–(A5) are satisfied Then for any u0 ∈ X = D(ϕ) ∩ D(ψ) there exists a unique strong solution u to (ACP), with norm bound u(t) X ≤ eγt u0 X , ∀ t ≥ 0, (4.4) 0,1/2 such that u ∈ Cloc (R+ ; X) In addition, solutions depend continuously on initial data: there exists two constants k3 ∈ R, k4 ≥ such that u(t) − v(t) X ≤ M (t, u0 , v0 )ek3 t u0 − v0 Here M (t, u0 , v0 ) := exp k4 e2γ+ t ( u0 X ∨ v0 ∀ u0 , v0 ∈ X X X) (4.5) REMARK 4.1 The coefficient on the right-hand side of (4.5) seems to be a little bit complicated So it may be meaningful to describe a particular case of (4.5) Suppose that γ ≤ 0, and introduce the ball BL := {v ∈ X; v X ≤ L} Setting U (t)u0 := u(t) and log M := k4 L2 , we have U (t)u0 − U (t)v0 X ≤ M ek3 t u0 − v0 X ∀ t ≥ 0, ∀ u0 , v0 ∈ BL This implies by (4.4) that {U (t)} forms a Lipschitz semigroup on the closed convex subset BL in the sense of Kobayashi-Tanaka [19] OUTLINE OF PROOFS Let u0 ∈ D(ϕ) ∩ D(ψ) and ε > Then under condition (A2) there exists a unique strong solution uε to (ACP)ε such that Copyright © 2006 Taylor & Francis Group, LLC Strong L2 -wellposedness in the complex Ginzburg-Landau equation 279 for all T > 0, uε ∈ C 0,1/2 ([0, T ]; X) ([29, Proposition 3.1(i)]) In addition we have uε (t) X ≤ eγt u0 X ∀ t ≥ 0, t t 2λ (4.6) ϕ(uε (s)) ds + qκ ψ(Jε uε (s)) ds ≤ 2γ+ t e u0 2 X, ∀ t ≥ 0; (4.7) note that Re ((κ + iβ)∂ψε (uε ), uε ) ≥ κRe (∂ψ(Jε uε ), Jε uε ) = qκψ(Jε uε ) (see [29, Lemma 3.2]) Next, (4.7) yields under conditions (A2) and (A3) that ϕ(uε (t)) ≤ eK(t,u0 ) ϕ(u0 ), t Suε (s) X ds k1 := 2γ+ + (1 − θ) c1 (4.8) K(t,u0 ) e ϕ(u0 ), λ ≤ Here K(t, u0 ) := k1 t + k2 e2γ+ t u0 ∀ t ≥ 0, X, ∀ t ≥ (4.9) with κ2 + β , k2 := θ c1 κ2 + β /(2qκ) Therefore we see from (4.6), (4.9), (A4) and (ACPeq)ε that {Suε }, {∂ψε (uε )}, {Dt uε } are bounded in L2 (0, T ; X) (4.10) As a consequence {uε } is equicontinuous on [0, T ]: uε (t) − uε (s) X ≤ Dt uε L2 (0,T ;X) |t − s|1/2 , ∀ t, s ∈ [0, T ] (4.11) In view of condition (A1) (4.8) yields that for every t ∈ [0, T ] the family {uε (t)}ε>0 is relatively compact in X Therefore, applying Ascoli’s theorem, we can find a function u ∈ C([0, ∞); X) as the limit of a convergent subsequence {uεn } selected from {uε } such that u = lim uεn in C([0, T ]; X), n→∞ ∀ T > 0, (4.12) with u(0) = u0 ∈ D(ϕ) ∩ D(ψ) Letting n → ∞ in (4.11) with ε replaced with εn , we have u ∈ C 0,1/2 ([0, T ]; X) for any T > Since uεn − Jεn uεn = εn ∂ψεn (uεn ), (4.10) yields that u = lim Jεn uεn in L2 (0, T ; X), n→∞ ∀ T > (4.13) Since S, Dt are weakly closed and ∂ψ is demi-closed, we see from (4.12), (4.13) and (4.10) that Suεn → Su, Dt uεn → Dt u and ∂ψεn (uεn ) = ∂ψ(Jεn uεn ) → ∂ψ(u) as n → ∞ weakly in L2 (0, T ; X) Therefore, we can conclude that u is a strong solution to (ACP) (for details see [29, Section 4]) Copyright © 2006 Taylor & Francis Group, LLC 280 N Okazawa Here we want to mention two inequalities for strong solutions u to (ACP): t 2λ t ϕ(u(s)) ds + qκ tϕ(u(t)) + ψ(u(s)) ds ≤ λ t s Su(s) X ds ≤ K(t, e 4λ u0 2γ+ t e u0 X )+2γ+ t u0 X, ∀ t ≥ 0, (4.14) X, ∀ t ≥ 0, (4.15) which will be essential in the proof of Theorem 4.2 To prove (4.15) begin with (4.1) with ∂ψε and Jε u, respectively, replaced with ∂ψ and u (by letting ε ↓ 0) (Then an error in the proof of (4.15) in [30] is easily corrected.) The important role of condition (A5) in Theorem 4.2 is to derive (4.5) as a consequence of (4.14) and to guarantee the uniqueness of strong solutions to (ACP) The constants kj (j = 3, 4) in (4.5) are given in almost the same form as for k1 and k2 : k3 := γ + (1 − θ) c3 κ2 + β , k4 := θ c3 κ2 + β /(2qκ) At the same time (4.5) enables us to extend the set of initial data from D(ϕ)∩ D(ψ) to its closure D(ϕ) ∩ D(ψ) = X The situation is completely similar to the inference in Theorem 3.2 In fact, (4.15) implies that Su L2 (T −1 ,T ;X) is bounded for all T > Finally, we shall verify (4.1) and (4.2) when S = ∂ϕ, B = ∂ψ are given by (2.1), (2.2), respectively First we state a formula for the derivative of the function Jε u by which we can carry out integration by parts in the computation of (Su, ∂ψε (u))L2 LEMMA 4.1 For ε ∈ [0, ∞) and x ∈ Ω put uε (x) := (Jε u)(x) = (1 + ε∂ψ)−1 u(x), u(x), ε > 0, ε = (a) ([29, Lemma 6.1 with p = 2]) If u ∈ H01 (Ω), then uε ∈ H01 (Ω) (as a function of x), with  q−2  ε > 0, |uε |q−4 uε Re(uε ∇x u), ∇x u − ε q−2 Jac (4.16) ∇x uε = + ε|uε |  ∇x u, ε = 0, where Jac := (1 + ε|uε |q−2 )(1 + ε(q − 1)|uε |q−2 ) is the Jacobian determinant (b) ([31]) If u ∈ D(∂ψ), then uε ∈ C ([0, E]; L2 (Ω)) ∀ E > (as a function of ε), with   − ∂ψε (u), ε > 0, ∂uε + ε(q − 1)|uε |q−2 =  ∂ε  −∂ψ(u), ε = Copyright © 2006 Taylor & Francis Group, LLC Strong L2 -wellposedness in the complex Ginzburg-Landau equation 281 For a proof of (a) put uε (x) = vε (x) + iwε (x) Then, applying the implicit (or inverse) function theorem to the simultaneous equation derived from the real and imaginary parts of uε + ε|uε |q−2 uε = u ∈ H01 (Ω) ∩ C (Ω), we can obtain the formulas for ∇x vε and ∇x wε Therefore, Jac appears when we solve the simultaneous equation for ∇x vε and ∇x wε In the second step the usual approximating procedure yields (4.16) Verification of (4.1) Put uε = Jε u as in Lemma 4.1 Then in view of the form of Yosida approximation ∂ψε (u) = ε−1 (u − uε ) we see from (4.16) that (Su, ∂ψε (u))L2 is written as Ω |uε |q−2 |∇u|2 dx + + ε|uε |q−2 Ω q−2 |uε |q−4 (uε ∇u) · Re (uε ∇u) dx, Jac (4.17) where ∇ := x Using Hăolders inequality, we have, for u H01 (Ω) ∩ H (Ω), (q − 1)−1 |(Su, ∂ψε (u))L2 | ≤ |uε |q−2 |∇u|2 dx ≤ uε Ω q−2 Lq ∇u Lq (4.18) In the next step we need the inequality ∇u Lq ≤ C ∇u 1−a L2 ∆u a L2 , ≤ a := N N ≤ − q (4.19) This is a consequence of the well known Gagliardo-Nirenberg interpolation inequality (see (4.22) below) Let θ be as defined by (4.3) Then, applying (4.19) and Young’s inequality, we can show that for any η > there exists C(η) > such that (4.1) holds: |(Su, ∂ψε (u))L2 | ≤ C(η)ψ(uε )θ S 1/2 u L2 + η Su L2 , uε = Jε u Thus we can explain the restriction on q: θ= N N q−2 ⇐⇒ q ≤ + ≤ − ≤ ⇐⇒ a = N q q (1 − a)q REMARK 4.2 It is worth noticing that (3.6) follows from (4.17) The proof is fairly simple (see [29, Lemma 6.2 with p = 2]) In fact, setting Ω |uε |q−2 |∇u|2 dx, + ε|uε |q−2 Ω |uε |q−4 (uε ∇u) · Re (uε ∇u) dx, Jac I1 (u) := I2 (u) := Copyright © 2006 Taylor & Francis Group, LLC 282 N Okazawa we see from (4.17) that the real and imaginary parts of (Su, ∂ψε (u))L2 have similar expressions: Re (Su, ∂ψε (u))L2 = I1 (u) + (q − 2)Re I2 (u) ≥ 0, (4.20) Im (Su, ∂ψε (u))L2 = (q − 2)Im I2 (u) (4.21) Applying the Cauchy-Schwarz inequality to I2 (u), we have |I2 (u)|2 ≤ I1 (u)Re I2 (u) which can be combined with (4.20) and (4.21) as follows: (q − 2)−2 |Im (Su, ∂ψε (u))L2 |2 = |Im I2 (u)|2 = |I2 (u)|2 − |Re I2 (u)|2 ≤ I1 (u)Re I2 (u) − [Re I2 (u)]2 = [Re (Su, ∂ψε (u))L2 − (q − 2)Re I2 (u)]Re I2 (u) − [Re I2 (u)]2 =√ ≤ Re (Su, ∂ψε (u))L2 q−1 q − [Re I2 (u)] − (q − 1)[Re I2 (u)]2 [Re (Su, ∂ψε (u))L2 ]2 (q − 1) In the last step we have used the inequality ab = 2(a/2)b ≤ (a/2)2 + b2 Since Re (Su, ∂ψε (u))L2 ≥ as pointed out in (4.20), we obtain (3.6) with X = L2 and tan ωq = cq Verification of (4.2) First we prepare an inequality asserting the local Lipschitz continuity of the mapping z → |z|q−2 z = ∂(|z|q /q): |z|q−2 z − |w|q−2 w ≤ (q − 1)|z − w| ≤ |sz + (1 − s)w|q−2 ds q q |z| + |w| 2 (q−2)/q ∀ z, w ∈ C (z = w); in the proof we have used the convexity of the function z → |z|q , which is a simple consequence of Hăolders inequality (1/q + 1/q = 1): |sz + (1 − s)w|q ≤ (s|z| + (1 − s)|w|)q = s1/q |z| · s1/q + (1 − s)1/q |w| · (1 − s)1/q ≤ (s|z|q + (1 − s)|w|q )q/q (s + (1 − s))q/q = s|z|q + (1 − s)|w|q Copyright © 2006 Taylor & Francis Group, LLC q Strong L2 -wellposedness in the complex Ginzburg-Landau equation 283 Then we can prove the inequality for u, v ∈ L2(q−1) (Ω) ∩ H01 (Ω) ⊂ Lq (Ω): (q − 1)−1 (|u|q−2 u − |v|q−2 v, u − v)L2 |u(x) − v(x)|2 ≤ Ω v ≤ u = q q ψ(u) + ψ(v) 2 q Lq + 1 |u(x)|q + |v(x)|q 2 q Lq (q−2)/q (q−2)/q u−v u−v (q−2)/q dx Lq Lq This is nothing but (4.18) (in Verification of (4.1)) with uε Lq = (qψ(uε ))1/q 1/q and ∇u Lq replaced with (q/2)(ψ(u)+ψ(v)) and u−v Lq , respectively From now on we can compute in the same way as in the proof of (4.1) to conclude (4.2); in this case we can apply the Gagliardo-Nirenberg inequality itself (see Tanabe [33, Section 3.4] or Giga-Giga [11, Section 6.1]): w Lq ≤C w 1−a L2 ∇w a L2 , ≤ a := N N ≤ − q (4.22) Therefore, the “θ” coincides with that in (4.1) This result seems to be parallel to the fact that (3.2) and (3.4) in Theorem 3.1 hold for the same tan ωq Development in the future As a concluding remark we want to mention the possibility to unify Theorems 3.2 and 4.2 (see Remark 1.1) Now we introduce the following seven conditions on S and ψ: (B1) ∃ q ∈ [2, ∞) such that ψ(ζu) = |ζ|q ψ(u) for u ∈ D(ψ) and ζ ∈ C with Re ζ > (B2) ∂ψ is sectorially-valued: ∃ cq > such that |Im(∂ψ(u)−∂ψ(v), u−v)| ≤ cq Re(∂ψ(u)−∂ψ(v), u−v) ∀ u, v ∈ D(∂ψ) (B3) For the same constant cq as in (B2), |Im(Su, ∂ψε (u))| ≤ cq Re(Su, ∂ψε (u)) ∀ u ∈ D(S) (B4) D(S) ⊂ D(∂ψ) and ∃ C1 > such that ∂ψ(u) ≤ C1 ( u + Su ) for u ∈ D(S) Copyright © 2006 Taylor & Francis Group, LLC 284 N Okazawa (B5) ∃ θ ∈ [0, 1]; ∀ η > ∃ C2 = C2 (η) > such that for u, v ∈ D(ϕ) ∩ D(ψ) and ε > 0, |Im(∂ψε (u) − ∂ψε (v), u − v)| ≤ ηϕ(u − v) + C2 ψ(u) + ψ(v) θ u − v (B6) ∀ η > ∃ C3 = C3 (η) > such that for u ∈ D(S) and ε > 0, |Im(Su, ∂ψε (u))| ≤ η Su + C3 ψ(u)θ ϕ(u), where θ ∈ [0, 1] is the same constant as in (B5) (B7) ∃ C4 > such that for u, v ∈ D(∂ψ) and ν, µ > 0, |Im(∂ψν (u) − ∂ψµ (u), v)| ≤ C4 |ν − µ| σ ∂ψ(u) + τ ∂ψ(v) , where σ, τ > are constants satisfying σ + τ = Using these conditions (partially or as a whole) we can assert THEOREM 5.1 ([31]) Let λ, κ ∈ R+ and α, β, γ ∈ R (I) (Accretive nonlinearity) Assume that |β|/κ ∈ [0, c−1 q ] and conditions (B1)− (B3) are satisfied Then for any initial value u0 ∈ D(ϕ) ∩ D(ψ) there exists a unique strong solution u ∈ C([0, ∞); X) to (ACP) such that 0,1 (a) u ∈ Cloc (R+ ; X), with u(t) ≤ eγt u0 ∀ t ≥ 0; (b) Su, ∂ψ(u), Dt u ∈ L∞ loc (R+ ; X); 1,1 (c) ϕ(u), ψ(u) ∈ Wloc (R+ ) Furthermore, let v be the unique strong solution to (ACP) with v(0) = v0 ∈ X Then u(t) − v(t) ≤ eγt u0 − v0 ∀ t ≥ (5.1) (II) (Nonaccretive nonlinearity) Assume that |β|/κ ∈ (c−1 q , ∞) and the seven conditions (B1)−(B7) are satisfied Then for any u0 ∈ X = D(S) there exists a unique strong solution u ∈ C([0, ∞); X) to (ACP) Also, u has property (c) in part (I) and 0,1/2 (a) u ∈ Cloc (R+ ; X), with u(t) ≤ eγt u0 (b) Su, ∂ψ(u), Dt u ∈ ∀ t ≥ 0; L2loc (R+ ; X) Furthermore, let v be the unique strong solution to (ACP) with v(0) = v0 ∈ X Then u(t) − v(t) ≤ eK1 t+K2 e 2γ+ t ( u0 ∨ v ) Copyright © 2006 Taylor & Francis Group, LLC u0 − v0 ∀ t ≥ 0, (5.2) Strong L2 -wellposedness in the complex Ginzburg-Landau equation 285 where the constants K1 and K2 depend on |β| − c−1 q κ > 0: K1 := γ + (1 − θ)(|β| − c−1 q κ)C2 , K2 := θ (|β| − c−1 q κ)C2 /(2qκ) (5.3) Part (I) is nothing but Theorem 3.2 (in view of Remark 3.1), while Part (II) generalizes Theorem 4.2 to the effect that the compactness of the level sets for ϕ is replaced with the new condition (B7) which guarantees, together with condition (B5), the convergence of the family of solutions to (ACP)ε ; the verification of (B7) is based on Lemma 4.1 (b) Therefore the boundedness of Ω is not required when we apply Theorem 5.1 Part (II) to problem (CGL) REMARK 5.1 It is observed from (5.3) that K1 → γ and K2 → as |β| → c−1 q κ, that is, (5.2) coincides with (5.1) in this limit Acknowledgments The author expresses his hearty thanks to the organizers of the meeting, “Evolution Equations: Inverse and Direct Problems,” especially Angelo Favini and Alfredo Lorenzi, for giving him a chance to contribute to this volume References [1] H Amann: Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, Differential Operators and Nonlinear Analysis (Friedrichroda, 1992), 9–126, TeubnerText Math., 123, Teubner, Stuttgart, 1993 [2] H Amann: Linear and Quasilinear Parabolic Problems, I: Abstract Linear Theory, Monographs in Math 89, Birkhăauser Verlag, Basel and Boston, 1995 [3] H Amann: On the strong solvability of the Navier-Stokes equations, J Math Fluid Mech (2000), 16–98 [4] I.S Aranson and L Kramer: The world of the complex Ginzburg-Landau equation, Reviews of Modern Physics 74 (2002), 99–143 [5] V Barbu: Nonlinear semigroups and differential equations in Banach spaces, Noordhoff International Publishing, Leyden, The Netherlands, 1976 [6] P Bechouche and A Jă ungel: Inviscid limits for the complex GinzburgLandau equation, Comm Math Phys 214 (2000), 201–226 Copyright © 2006 Taylor & Francis Group, LLC 286 N Okazawa [7] H Br´ezis: Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, Mathematics Studies 5, North-Holland, Amsterdam, 1973 [8] H Br´ezis: Analyse Fonctionnelle, th´eorie et applications, Masson, Paris, 1983 [9] H Br´ezis and A Friedman: Nonlinear parabolic equations involving measures as initial conditions, J Math Pures et Appl 62 (1983), 73–97 [10] V.V Chepyzhov and M.I Vishik: Attractors for equations of Mathematical Physics, Colloquium Publications 49, Amer Math Soc., Providence, 2002 [11] Y Giga and M Giga: Nonlinear partial differential equations, asymptotic behavior of solutions and self-similar solutions, Kyoritsu Shuppan, Tokyo, 1999 (in Japanese); 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