**Courant Lecture Notes**

Volume: 10;
2003;
323 pp;
Softcover

MSC: Primary 35;

Print ISBN: 978-0-8218-3399-5

Product Code: CLN/10

List Price: $49.00

Individual Member Price: $39.20

**Electronic ISBN: 978-1-4704-1760-4
Product Code: CLN/10.E**

List Price: $49.00

Individual Member Price: $39.20

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#### Supplemental Materials

# Semilinear Schrödinger Equations

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*Thierry Cazenave*

A co-publication of the AMS and the Courant Institute of Mathematical Sciences at New York University

The nonlinear Schrödinger equation has received a great deal of
attention from mathematicians, particularly because of its applications to
nonlinear optics. It is also a good model dispersive equation, since it is
often technically simpler than other dispersive equations, such as the wave or
the Korteweg-de Vries equation. From the mathematical point of view,
Schrödinger's equation is a delicate problem, possessing a mixture of the
properties of parabolic and elliptic equations. Useful tools in studying the
nonlinear Schrödinger equation are energy and Strichartz's estimates.

This book presents various mathematical aspects of the nonlinear
Schrödinger equation. It studies both problems of local nature (local
existence of solutions, uniqueness, regularity, smoothing effect) and problems
of global nature (finite-time blowup, global existence, asymptotic behavior of
solutions). In principle, the methods presented apply to a large class of
dispersive semilinear equations. The first chapter recalls basic notions of
functional analysis (Fourier transform, Sobolev spaces, etc.). Otherwise, the
book is mostly self-contained.

It is suitable for graduate students and research mathematicians interested
in nonlinear partial differential equations and applications to mathematical
physics.

Titles in this series are co-published with the Courant Institute of Mathematical Sciences at New York University.

#### Table of Contents

# Table of Contents

## Semilinear Schrodinger Equations

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- Notation ix10
- Preliminaries 116
- The linear Schrödinger equation 2944
- The Cauchy problem in a general domain 5570
- The local Cauchy problem 8398
- Regularity and the smoothing effect 147162
- Global existence and finite-time blowup 163178
- Asymptotic behavior in the repulsive case 211226
- Stability of bound states in the attractive case 255270
- Further results 283298
- Bibliography 305320
- Back Cover Back Cover1346

#### Readership

Graduate students and research mathematicians interested in nonlinear partial differential equations and applications to mathematical physics.

#### Reviews

The book, written by one of the leading expert on the subject, is also an up-to- date source of references for recent results and open problems, well represented in its extensive bibliography. It can certainly be used as a guideline for a course to an audience with a sufficient background on functional analysis and partial differential equations.

-- Zentralblatt MATH

In summary, the author gives a well balanced treatment of the many types of mathematical results... This book would be an excellent place to start for readers interested in an introduction to these topics. There is an extensive bibliography which nicely complements the author's discussions.

-- Woodford W. Zachary for Mathematical Reviews