Abstract Non Linear Wave Equations
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507
Michael Reed
Abstract Non Linear Wave Equations
Springer-Verlag Berlin. Heidelberg. New York 197 6
Author Michael C. Reed Department of Mathematics Duke University Durham, North Carolina 27706 USA
Library of Congress Cataloging in Publieation Data
Reed, Michael. Abstract non-linear wave equations. (Lecture notes in mathematics ; 507) Based on lectures delivered at the Zentz~m f~r interdisziplin~me Forsehung in 1975. Bibliography: p. 1. Wave equation. I. Title. II. Series : Lecture notes in mathematics (Berlin) ; 507. QA3.La8 no. 507 [QCI74.26.W3] 510'.8s [530.1'24] 76-2551
AMS Subject Classifications (1970): 35L60, 47H15 ISBN 3-540-07617-4 ISBN 0-387-07617-4
Springer-Verlag Berlin Heidelberg 9 New 9 York Springer-Verlag New Y o r k . Heidelberg Berlin 9
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under w 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. 9 bySpringer-Verlag Berlin Heidelberg 9 1976 Printed in Germany Printing and binding: Beltz, Offsetdruck, Hemsbach/Bergstr.
Preface
These notes cover a set of eighteen lectures delivered at th e Zentrum f~r interdisziplin~re Forschung of the University of Bielefeld in the summer of 1975 as part of the year long project "Mathematical Problems of Quantum Dynamics".
It is a pleasure to thank the Zentrum
for the opportunity to give these lectures and the Physics faculty of the University of Bielefeld for their warmth and hospitality. people deserve special %hanks:L. C. Pfister,
Streit,
Three
for extending the invitation,
for help in the preparation of the man~scrlpt,
and M. K~mper
for her excellent typing.
Mike Reed Bielefeld, August,
1975
Table
Introduction
Chapter
1
I. L o c a l
of Contents
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Existence and global
and properties existence
1
of s o l u t i o n s
. . ~ . . . . . . . . . . . . . . . .
5
2. A p p l i c a t i o n s : A.
utt
-
~u + m 2 u
B.
The
case
C.
Other
D.
The
E.
An example
F.
The
p,
7.
8. S c a t t e r i n g 9. G l o b a l
,
D
=
3 .
.
.
.
.
.
.
.
.
.
II
20
equation global
. . . . . . . . . . . . . . . . . . existence
fails
and K l e i n - G o r d o n
. . . . . . . . . . .
equations
. . . . . . . .
. . . . . . . . . . . . . . . . . . . . .
speed
and continuous
25 29 33 40
dependence 49
. . . . . . . . . . . . . . . . . . . . . . . . .
56
. . . . . . . . . . . . . . . . . . . . . . . . . . .
64
theory
of scattering for s m a l l
existence
10.Existence
3
A . . . . . . . . . . . . . . . . . . . . . .
Dirac
Scattering
Formulation
=
. . . . . . . . . . . . . . . . . . . . . . . . . . .
solutions
2
n
19
where
propagation
,
. .
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