Dynamics of Infinite Dimensional Systems

508 Pages·1987·13.248 MB·English

by  Herbert Amann (auth.), Shui-Nee Chow, Jack K. Hale (eds.)

Dynamics of Infinite Dimensional Systems
NATO ASI Series
Advanced Science Institutes Series
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Commlttee, which aims at the dissemination of advanced scientific and
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E Applied Sciences
F Computer and Springer-Verlag
Systems Sciences Berlin Heidelberg New York
G Ecological Sciences London Paris Tokyo
H Cell Biology
Series F: Computer and Systems Sciences Vol. 37
Dynamics of Infinite Dimensional
Systems
Edited by
Shui-Nee Chow
Department of Mathematics
Michigan State University
East Lansing, MI 48824-1027
U.SA
Jack K. Haie
Division of Applied Mathematics
Brown University
Providence, RI 02912
U.SA
Springer -Verlag
Berlin Heidelberg New York London Paris Tokyo
Published in cooperation with NATO Scientific Affairs Division
Proceedings of the NATO Advanced Study Institute on Dynamics of Infinite Dimen
sional Systems, held in Lisbon, Portugal, May 19-24, 1986
ISBN 978-3-642-86460-5 ISBN 978-3-642-86458-2 (eBook)
DOI 10.1007/978-3-642-86458-2
Library of Congress Cataloging in Publication Data. NATO Advanced Study Institute on Dynamics of
Infinite Dimensional Systems (1986: Lisbon, Portugal) Dynamics of infinite dimensional systems. (NATO
ASI series. Series F, computer and systems sciences; vol. 37) "Proceedings of the NATO Advanced Study
Institute on Dynamics of Infinite Dimensional Systems held in Lisbon, Portugal, May 19-24, 1986"-Tp.
verso. 1. Differential equations-Congresses. 2. Differential equations, Partial-Congresses. I. Haie, Jack
K. 11. Chow, Shui-Nee. 111. Title. IV. Series. QA372.N38 1986515.3'587-26383
ISBN 978-3-642-86460-5
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© Springer-Verlag Berlin Heidelberg 1987
Softcover reprint of the hardcover 1 st edition 1987
2145/3140-543210
PREFACE
The 1986 NATO Advanced Study Insti tute on Dynamics of Infini te
Dimensional Systems was held at the Instituto Superior Tecnico. Lisbon.
Portugal.
In recent years. there have been several research workers who have
been considering partial differential equations and functional
differential equations as dynamical systems on function spaces. Such
approaches have led to the formulation of more theoretical problems that
need to be investigated. In the applications. the theoretical ideas have
contributed significantly to a better understanding of phenomena that
have been experimentally and computationally observed.
The investigators of this development come wi th several different
backgrounds - some from classical partial differential equations. some
from classical ordinary differential equations and some interested in
specific applications. Each group has special ideas and often these
ideas have not been transmitted from one group to another.
The purpose of this NATO Workshop was to bring together research
workers from these various areas. It provided asoundboard for the
impact of the ideas of each respective discipline. We believe that goal
was accomplished. but time will be a better judge.
We have included the list of participants at the workshop. with most
of these giving a presentation. Although the proceedings do not include
all of the presentations. it is a good representative sampie.
We wish to express our gratitude to NATO. and.to Dr. M. di Lullo of
NATO. who unfortunately did not live to see the completion of this
project.
We also greatly appreciate the additional financial support from the
Uni ted States Air Force Office of Scientific Research. the Uni ted States
Army Research Office. the Instituto Nacional de Investiga~ao Cientifica
(Portugal) and the Junta Nacional de Investigayao Cientifica (Portugal).
We would also like to thank the Secretary of State for Higher
Education. Professor Fernando Ferreira Leal. the Secretary of State for
Scientific Research. Professor Arantes e Oliveira. and the President of
the Executive Council and Scientific Council of the Insti tuto Superior
Tecnico. Professor Diamantino Durao.
For hosting the conference. we wish to thank the CompI exo
Interdisciplinari of the Instituto Nacional de Investigarao Cientifica.
July 1987 Shui-Nee Chow. East Lansing
Jack K. HaIe. Providence
Table of Oontents
Amann, H.
Semilinear Parabolic Systems Under Nonlinear Boundary Conditions ..•. 1
Angenent, S.B.
The Shadowing Lemma for Elliptic PDE . • • • • • • • • • • • • • • • • • 7
Ball, J.M. and Carr J.
Coagulation-Fragmentation PYnamics ................. 23
Becker, L., Bur ton , T.A, and Zhang, S.
Functional Differential Equations and Jensen's Inequality ....... 31
Canada, A.
Method of Upper and Lower Solutions for Nonlinear Integral Equations and an
Application to an Infectious Disease Model . . . . . . . . . . . . • . . 39
Chossat, P., Demay, Y., and Iooss, G.
Competi ti on of Azimuthal Modes and Quasi-Periodic Flows in the
Couette-Taylor Problem .. ..... . . 45
Chow, S-N. and Lauterbach, R.
On Bifurcation for Variational Problems . . . . . . . . . . . . . • . . 57
Cushman, R. and Sanders, J.A.
Nilpotent Normal Form in Dimension 4 . . . . . . . . . . . . . . . . . . 61
Diekmann, O.
Perturbed Dual Semigroups and Delay Equations . . . . . . . . . . . . . 67
Ferreira, J.M.
On Operators Which Leave Invariant' a Half-Space . • . . . . . . . . . . 75
Fiedler, B.
Global Hopf Bifurcation in Reaction Diffusion Systems with Symmetry .. 81
Fitzgibbon, W.E.
Longtime Behavior for a Class of Abstract Integrodifferential Equations 105
Fusco, G.
Describing the Flow on the Attractor of One Dimensional Reaction Diffusion
Equations by Systems of ODE . . . . . . . • . . . . . . . . . . . . . 113
HaIe, J.K.
Asymptotic Behavior of Gradient Dissipative Systems 123
Henry, D.B.
Generic Properties of Equilibrium Solutions by Perturbation of the Boundary
129
Lune!, S.M. V.
Complex Analytical Methods in RFDE Theory 141
VIII
Mawhin. J.
Quali tative Behavior of the Solutions of Periodic First Order Scalar
Differential Equations with Strictly Convex Coercive Nonlinearity 151
Magalhäes. L. T .
The Spectrum of Invariant Sets for Dissipative Semiflows . . . . . 161
Marcati. P.
Approximate Solutions to Conservation Laws via Convective Parabolic
Equations: Analytical and Numerical Results .... 179
Mischaikow. K.
Conley's Connection Matrix 179
Mora. X. and Sola-Morales J.
Existence and Non-Existence of Finite-Dimensional Globally Attracting
Invariant Manifolds in Semilinear Damped Wave Equations . . . . 187
Nishiura. Y. and Fujii. H.
SLEP Method to the Stability of Singularly Perturbed Solutions with
Multiple Internal Transition Layers in Reaction-Diffusion Systems 211
Nussbaum. R.D.
Iterated Nonlinear Maps and Hilbert's Projective Metric: A Summary 231
Oliva. W.M.
Jacobi Matrices and Transversality 249
Rocha. c.
Examples of Attractors in Scalar Reaction-Diffusion Equations 257
Sattinger. D.H. and Zurkowski. V.D.
Gauge Theory of Backlund Transformations. I 273
Souganidis. P.E.
Recent Developments in the Theory of Nonlinear Scalar First and Second
Order Partial Differential Equations . . . . . . . . . . . . . . .. 301
Staffans. o.
Hopf Bifurcation for an Infinite Delay Functional Equation . . .. 313
Stech. H.W.
A Numerical Analysis of the Structure of Periodic Orbi ts in Autonomous
Functional Differential Equations . . . . . . . . • . 325
Tartar. L.
Oscillations and Asymptotic Behaviour for two Semilinear Hyperbolic Systems
341
Terman. D.
An Application of the Conley Index to Combustion . . 357
Ulrich. K.
Path Continuation - A Sensitivity Analysis Approach 373
IX
Ushiki, S. and Lozi, R.
Confinor and Anti-confinor in Constrained "Lorenz" System 385
Vanderbauwhede, A.
Invariant Manifolds in Infinite Dimensions. . . . . . . . . . . . . . 409
van Moerbeke, P.
Linearizing Completely Integrable Systems on Complex Algebraic Tori . . 421
Vegas, J.M.
On Some Dynamical Aspects of Parabolic Equations with Variable Domain . 451
Wal ther, H-D.
Bifurcation from Homoclinic to Periodic Solutions by an Inclination Lemma
with Pointwise Estimate . . . . . . . . . . . . . . . . . . . . . 459
Williamson, F.
Approximate Methods for Set Valued Differential Equations with Delays . 471
Wisniewski, H.S.
Bounds for the Chaotic Behavior of Newton's Method 481
List of Participants . 511
Semilinear Parabolic Systems Under
Nonlinear Boundary Conditions
Herbert Amann
Mathematisches Institut
Universität Zürich
Rämistrasse 74
CH-8001 Zürich
We study problems of the form
(A) u+Au=F(u), Bu=G(u), u(O) = u
o
which can be considered as abstract counterparts to semilin
ear parabolic systems under non linear boundary conditions.
Typica1 examples, to which our abstract theory applies, are of
the form
ot u - 0/ a j k 0ku) f(x,u,ou) in n x (0,00) ,
(P) ajk)oku g(x,u) on on x (0,00)
u ( • ,0) u on n ,
0
n 1 n
where nc JR is a bounded smooth domain, v = (v , ••• ,v ) is the
out ern 0 rma Ion 0 n an d u = (u 1 , •.. , u N) isa n N- v e c tor val u e d
function. We assume that (P) represents a parabo1ic system and
fand gare smooth functions such that f satisfies some poly
nomial growth restriction with respect to ou. Instead of
giving the precise assumptions we mention onLy that all as-
NATO ASI Series, Vol. F37
Dynamics of Infinite Dimensional Systems
Edited by S.-N. Chow, and J.K. Haie
© Springer-Verlag Berlin Heidelberg 199?
2
sumptions ars satisfisd if (P) is of ths vsry spscial form
ot U 1 - C4 11l'1 U1 - a. 12uA U 2 f1(U1,U2)
in n x (0,00) ,
ot U 2 - C4 211'1 U1 - a. 22uA U 2 f2(u2,u2)
11 ou 1 1 1 2
ov g (u ,u )
C4 on an x (0,00) ,
21 ou 1 212
a. ov g (u ,u )
whsrs f1, f2, g1, gZ ars arbitrary smooth functions, providsd
11 22 . 11 22 12 21 2 21
a. >0, a. >0 and slthsr 4a. a. > (a. +a. ) or a. =0.
As for problsm (A) ws assums that W1, Wand OW1 ars
1
Banach spacss such that W c~W, whsrs c~ msans continuous in-
jsction. Morsovsr ws assums (for simplicity) that
(A, B) EIs om (W 1 ,W x aw 1 )
1 1 1
Thsn ws put WA:= ksrA and WB:= ksrlB), ws 1st A:=AIWB and
ws assums that
-A is ths infinitssimal gsnsrator of a
strongly continuous analytic ssmigroup
on W.
It follows that
Simpls hsuristic argumsnts lsad thsn to ths following "vari
ation of constants formula" for problsm (A):
-tA ft -(t-T)A
(1) u(t)=s u + s (F(U(T))+AR1GCU(T)))dT,O:st<OO.
o 0
For a dstailsd dsscription of ths dsduction of this formula ws