**Semigroups
for Delay Equations**

**András Bátkai and Susanna Piazzera**

**A. K. Peters: Wellesley MA, 2005,**

**Research Notes in
Mathematics vol. 10, **

**ISBN****:
1-56881-243-4.**

The book presents in a systematic way how delay equations can
be studied in L^{p}-history spaces. Using recent operator semigroup
methods the existence of solutions and the asymptotic behavior is analyzed in
detail. Main emphasis is laid on uniform exponential stability and hyperbolicity
of the solution semigroup, but other properties such as positivity, almost
periodicity or non-uniform exponential stability are also considered. The
efficiency of the theoretical results is demonstrated on various examples.

To make the book self-contained, the needed operator semigroup theoretic results are collected in the first two chapters. Then the well-posedness of the delay equations is showed using perturbation theoretic arguments.

The asymptotic behavior of the solutions is first analyzed for regular equations using spectral mapping theorems and characteristic equations. This is extended to the non-regular case using resolvent growth estimates and perturbation arguments. As an important application, the problem of small delays is also considered.

Finally, the theory is extended to wave equations with delay and to parabolic equations with delays in the highest order derivatives.

Some less well-known results on vector valued integration and Sobolev spaces are presented in the appendix.

An extensive bibliography and a detailed index may help to use the book as a reference tool.

You find the homepage of the publisher here. The official flyer of the book is here (PDF-file). In Europe, you may also want to consider Amazon.de.

See the entry in Zentralblatt Math database (PDF-format).

See the entry in Mathematical Reviews (PDF-format)

A Review in the Newsletter of the European Mathematical Society

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IMA Journal of
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Fuhrman, M., Tessitore, G., *Generalized directional
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Appl. Math. Optim. **51** (2005), 279–332.

Sikolya, E., *Flows in networks with dynamic ramification
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Hadd, S., Idrissi, A., *Regular linear systems governed by
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Errata