##
**Theory and applications of partial functional differential equations.**
*(English)*
Zbl 0870.35116

Applied Mathematical Sciences. 119. New York, NY: Springer. x, 429 p. (1996).

The main subject of this book is the qualitative behavior of the solutions of semilinear partial differential equations with time delay and its applications to problems arising from physics, chemistry and biology. For many decades the powerful methods of semigroup theory have been widely used also in applied mathematics to treat partial differential equations of evolutionary type as ordinary differential equations in Banach spaces by studying the so-called abstract form of the partial differential equation. In the present book, this is written as:
\[
u'(t)= Au(t)+ F(u_t),\quad t\geq 0,
\]
where \(A\) is the infinitesimal generator of a strongly continuous (or possibly analytic) semigroup of bounded linear operators in a Banach space \(X\) of functions (of space variables), \(F\) is a nonlinear function from \(C(-r,0;X)\) to \(X\) and \(u_t\) is defined as \(u_t(s)= u(t+s)\) for \(s(-r,0)\). In this way, it is possible to use the semigroup theory applied to abstract evolution equations to generalize (when it is possible) to the infinite-dimensional case many techniques and results of ordinary differential equations. In fact, after some preliminaries and a collection of results from the classical semigroup theory and elliptic equations, the author treats a great number of subjects relating the asymptotic and qualitative behavior of the solution of a class of semilinear equations with delay: compactness of solution semiflows, decomposition of state spaces by invariant subspaces, linearized stability, periodic solutions, Hopf bifurcations, influence of diffusivity on the stability, upper and lower solutions, travelling wave solutions.

A remarkable quality of the book is the abundance of examples and applications to mathematical models from population dynamics, cellular biology, control theory, physics, meteorology, viscoelasticity, and many other fields. They are exposed with great detail and numerous bibliographical references are given (26 pages in total). They can convince also the applied mathematician about the utility of the functional analytic approach to partial differential equations.

A remarkable quality of the book is the abundance of examples and applications to mathematical models from population dynamics, cellular biology, control theory, physics, meteorology, viscoelasticity, and many other fields. They are exposed with great detail and numerous bibliographical references are given (26 pages in total). They can convince also the applied mathematician about the utility of the functional analytic approach to partial differential equations.

Reviewer: G.Di Blasio (Roma)

### MSC:

35R10 | Partial functional-differential equations |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

92B20 | Neural networks for/in biological studies, artificial life and related topics |

37N99 | Applications of dynamical systems |

92D25 | Population dynamics (general) |