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Book review

Author(s) Dore, Giovanni (ed.)
Favini, Angelo (ed.)
Obrecht, Enrico (ed.)
Venni, Alberto (ed.)
Publisher Marcel Dekker
Year of publication 1993
Reviewed by P. Zabrejko

This book is Proceedings of the Second Bologna Conference on Differential Equations in Banach Spaces. The main directions that were reflected in scientific program of this meeting as well as in the book were not only abstract evolutionary differential equations but also those branches of functional analysis that have proven to be useful for studying of evolutionary equations and so the book covers such fields as semigroups theory, functional calculus, perturbation theory, interpolation and extrapolation spaces, spaces of vector-valued functions, Volterra operator equations and so on. A lot of talkings included in this book are concerned with applied aspects of the evolutionary differential equations such as Karman systems, semilinear wave equations, Euler-Bernoulli and Kirchhof equations and others.

In general the book contains 19 articles. The article by Paolo Acquistapace Abstract Linear Nonautonomous Parabolic Equations: A Survey is devoted to the Cauchy problem <formula> with a family of generators of analytic semigroups in a Banach space E; the author's approach is based on the investigation of the integral equation for function <formula> in cases when B(t,s) is equal to o of the operator-functions A(s), A(t), A(0) and <formula>; one can find numerous existence results and the description of different properties for corresponding evolutionary operators for all these cases as well as some illusttrative examples. The article by Marco Luigi Bernardi and Fabio Luterotti On Some Classes of Singular Variational Inequalities is devoted to the evolutionary variational inequalities with degenerating coefficients of parabolic type and in which the unilateral constraints concern the unknown function or the time derivative of the unknown function; for such inequalities the authors present some existence and perturbation results (some analogous results for variational inequalities of hyperbolic type were earlier obtained by M.L. Bernardi - P. Luterotti - G.A. Pozzi). In the article by Julio E. Bouillet Nonuniqueness in <formula> there is described the equation <formula> for which there exist two different in <formula> solutions <formula> and <formula> on <formula> satisfying to the name initial condition: <formula>

The article by Piermarco Cannarsa and Ciuseppe D. Prato Some Results on Abstract Evolution Equations of Hyperbolic Type deals with a new generalization of the well-known T. Kato theorem on fundamental solutions for evolution equations of hyperbolic type that allows to cover the Cauchy problem for the Kolmogoroff equation: <formula>

The article by Gabriella Di Blasio Interpolation and Extrapolation Spaces and Parabolic Equations deals with the Cauchy problem <formula> with a generator A of an analytic semigroup S(t) in a Banach space X; the author's approach is based on studying of this problem in the scale of Banach spaces D(O,p) that are defined as natural completenesses of linear subspaces of elements x from X for which have a sense the norm <formula> some new regularity results and existence of generalized solution results are given.

The article by Klaus-Jochen Engel On the Diagonalization of Certain Operator Matrices Related to Volterra Equations is devoted to the Cauchy problem<formula> in a Banach space X; this problem is reduced to study of some semigroup of continuous operators T(t) in a suitable function space; the main results are concerned with calculation of the spectrum for the generator <formula> of the semigroup T(t).

In the article by A. Favini and I. Lasiecka Second Order Abstract Equations with Nonlinear Boundary Conditions: Applications to a von Karman System with Boundary Damping the Cauchy problem <formula>; this system is a mathematical model for a lot of equations of mathematical physics, in particular nonlinear wave equations, Von Karnan plate equations, nonlinear Euler Bernoulli and Kirchhof plates equations and etc.; the main goal of this article is to present "theory" of wellposedness for the abstract system above and so the local and global existence and different regularity results are given.

The article by A. Favini and Hiroki Tanabe linear Parabolic Differential equations of Higher Order in Time deals with a generalization of the classical result of T. Kato - H. Tanabe on the linear parabolic equation of the third order <formula> with variable coefficients.

The article by A. Favini and R. Trigiani Analytic and Gevrey Class Semigroups Generated by A + iB and applications presents some new conditions under that the operator A + iB is a generator of a strongly continuous and nonexpansive semigroup of Gevrey class of some order; these results are applied to optimal control problems and inhomogeneous problems in Gevrey classes.

In the article by Jerome A. Goldstein The Kompaneets Equation is discussed some regularity problems for the classical Kompaneets system that arises in plasma physics and that is a spatially degenerated nonlinear parabolic partial differential equation; the main result is the description of a space in which the corresponding operator turns to be dissipative.

The article by Albrecht Holderrieth Multiplicative Perturbation of Resolvent Positive Operators presents a sufficient conditions under that the operator BA with a resolvent positive operator A is a resolvent positive operator too and in addition is a generator of strongly continuous semigroup.

The article by I. Lasiecka and D. Tataru Uniform Decay Rates for Semilinear Wave Equations with Nonlinear and Nonmonotone Boundary Feedbacks, without Geometric Conditions deals with the nonlinear system <formula> (<formula> is a bounded domain with a smooth boundary in <formula>); the main results are theorem on the existence of solution of this system on the <formula> in a suitable space and a theorem on asymptotic properties of these solutions.

The article by I. Lasiecka and R. Triggiani Sharp Trace Estimates of Solutions to Kirchhoff and Euler-Bernoulli Equations describes some curious tangential estimates for Kirchhoff-type and Euler-Bernoulli-type partial differential equations of the spatially fourth order; as application the phenomenon of elamination of geometrical conditions in the theories of exact controllability and uniform stabilization for plate equations is described.

The article by Ralph deLaubenfels Boundary Values of Holomorphic Semigroups, <formula> Functional Calculi, and the Inhomogeneous Abstract Cauchy Problem deals with the "regularization" problem for the (usually) unbounded group of operators <formula> where <formula> is a bounded strongly continuous holomorphic semigroup, in particular for the group of fractional powers of a fixed operator; as a result the <formula> functional calculus and some results on the inhomogeneous Cauchy problem are described.

The article by Jan Prüss Stability of Linear Evolutionary Systems with Applications to Viscoelasticity presents a description of linear equations of the viscoelasticity theory in details and then the existence, regularity and stability results for these equations; the results presented in this article cover some nonclassical media.

The article by Vincenzo Vespri Generation of Analytic Semigroups by Variational Operators with <formula> Coefficients describes the conditions under that an elliptic operator with <formula> coefficients generates an analytical semigroup in a suitable function space.

The article by G.F. Webb Asynchronous Exponential Growth in Differential Equations with Homogeneous Nonlinearities is devoted to asymptotic properties of solutions of abstract semilinear differential equations with homogeneous nonlinearities of form <formula> (A is a generator of a strongly continuous semigroup of positive linear operators in a Banach lattice X); the main result describes conditions under that there exists a nonzero operator <formula> for a eigen value <formula> of the homogeneous operator <formula>; in addition to the abstract result some illustrative examples from the population theory are given.

The article by L. Weis Inversion of the Vector-Valued Laplace Transform In <formula>-Spaces is devoted to the description of the precise class of Banach spaces for which the classical inversion theorem holds; a result on the range of the vector-valued Fourier transform and a Paley-Wiener analog for the Pettis norm and some applications to semigroup theory and the abstract Cauchy problem are represented too.

In the article by Atsushi Yagi Some Quasilinear Parabolic Problems In Applied Mathematics is devoted to studying a strongly coupled parabolic system that was introduced by N. Shigesada, K. Kawasaki and E. Teramoto to describe the population dynamics of two competitive species; the main results are the theorem on positivity of real solutions to this system, the local existence and uniqueness theorem and the global existence theorem.

As one may see the book contains reach and up-to-date information on the abstract and applied theory of linear differential equations in Banach spaces and its applications to some concrete problems of mathematical natural sciences. So this book is useful for all active investigators in field and as well as all mathematicians who are interested in theory evolution differential equations.