Semigroups for Delay Equations

by Batkai; Andras

• ISBN13: 9781568812434
• ISBN10: 1568812434
• Format: Hardcover
• Copyright: 2005-05-09
• Publisher: A. K. Peters

Summary

In most physical, chemical, biological and economic phenomena it is quite natural to assume that the system not only depends on the present state but also on past occurrences. These circumstances are mathematically described by partial differential equations with delay. This book presents, in a systematic fashion, how delay equations can be studied in Lp-history spaces. Appendices offering supplementary information and a comprehensive index make this book an ideal introduction and research tool for mathematicians, chemists, biologists and economists.

Author Biography

Andrßs Bßtkai earned his Ph.D. from the University of Tnbingen, Germany in 2000. He is Assistant Professor at the E÷tv÷s Lorand University, Budapest and winner of the 2003 Gyula Farkas Prize of the Jßnos Bolyai Mathematical Society.Susanna Piazzera earned her Ph.D. from the University of Tnbingen, Germany in 1999 and continues her research there as a Margarete von Wrangell fellow.

Table of Contents

Preface	ix
I Preliminary Results in Semigroup Theory	1	(40)
1 Semigroup Theory	3	(24)
1.1 Strongly Continuous Semigroups	3	(6)
1.2 Abstract Cauchy Problems	9	(2)
1.3 Special Classes of Semigroups	11	(5)
1.4 Perturbation Theory	16	(3)
1.5 Regularity Properties of Perturbed Semigroups	19	(6)
1.6 Notes and References	25	(2)
2 Spectral Theory and Asymptotics of Semigroups	27	(14)
2.1 Spectrum, Stability, and Hyperbolicity	27	(8)
2.2 Gearhart's Theorem	35	(3)
2.3 Notes and References	38	(3)
II Well-Posedness	41	(36)
3 The Delay Semigroup	43	(34)
3.1 The Semigroup Approach	43	(12)
3.2 Spectral Theory for Delay Equations	55	(5)
3.3 Well-Posedness: Bounded Operators in the Delay Term	60	(11)
3.4 Well-Posedness: Unbounded Operators in the Delay Term	71	(4)
3.5 Notes and References	75	(2)
III Asymptotic Behavior	77	(96)
4 Stability via Spectral Properties	79	(22)
4.1 Regularity of the Delay Semigroup	80	(6)
4.2 Stability via Spectral Mapping Theorem	86	(7)
4.3 The Critical Spectrum	93	(2)
4.4 Stability via Critical Spectrum	95	(4)
4.5 Notes and References	99	(2)
5 Stability via Perturbation	101	(24)
5.1 Stability and Hyperbolicity via Gearhart's Theorem	101	(14)
5.2 Fourier Multipliers	115	(1)
5.3 Hyperbolicity via Fourier Multipliers	116	(8)
5.4 Notes and References	124	(1)
6 Stability via Positivity	125	(20)
6.1 Positive Sernigroups	125	(2)
6.2 Stability via Positivity	127	(10)
6.3 The Modulus Semigroup	137	(6)
6.4 Notes and References	143	(2)
7 Small Delays	145	(16)
7.1 The Effect of Small Delays	145	(14)
7.2 Notes and References	159	(2)
8 More Asymptotic Properties	161	(12)
8.1 Asymptotic Properties of Perturbed Sernigroups	161	(5)
8.2 Asymptotic Properties of the Delay Semigroup	166	(4)
8.3 Notes and References	170	(3)
IV More Delay Equations	173	(60)
9 Second-Order Cauchy Problems with Delay	175	(34)
9.1 Dissipative Wave Equations in a Hilbert Space	175	(6)
9.2 Uniform Exponential Stability	181	(9)
9.3 Wave Equations with Bounded Delay Operators	190	(6)
9.4 Wave Equations with Unbounded Delay Operators	196	(10)
9.5 Notes and References	206	(3)
10 Delays in the Highest-Order Derivatives	209	(24)
10.1 The Perturbation Theorem of Weiss-Staffans	209	(2)
10.2 Well-Posedness	211	(11)
10.3 Spectral Theory	222	(4)
10.4 Stability and Hyperbolicity	226	(4)
10.5 Notes	230	(3)
Appendix—Vector-Valued Functions	233	(4)
A.1 The Bochner Integral	233	(2)
A.2 Vector-Valued Sobolev Spaces	235	(2)
Bibliography	237	(17)
Index	254