Synopsis :
I Equations of Scalar Type.- 1 Resolvents.- 1.1 Well-posedness and Resolvents.- 1.2 Inhomogeneous Equations.- 1.3 Necessary Conditions for Well-posedness.- 1.4 Perturbed Equations.- 1.5 The Generation Theorem.- 1.6 Integral Resolvents.- 1.7 Comments.- 2 Analytic Resolvents.- 2.1 Definition and First Properties.- 2.2 Generation of Analytic Resolvents.- 2.3 Examples.- 2.4 Spatial Regularity.- 2.5 Perturbed Equations.- 2.6 Maximal Regularity.- 2.7 Comments.- 3 Parabolic Equations.- 3.1 Parabolicity.- 3.2 Regular Kernels.- 3.3 Resolvents for Parabolic Equations.- 3.4 Perturbations.- 3.5 Maximal Regularity.- 3.6 A Representation Formula.- 3.7 Comments.- Appendix: k-monotone Kernels.- 4 Subordination.- 4.1 Bernstein Functions.- 4.2 Completely Positive Kernels.- 4.3 The Subordination Principle.- 4.4 Equations with Completely Positive Kernels.- 4.5 Propagation Functions.- 4.6 Structure of Subordinated Resolvents.- 4.7 Comments.- Appendix: Some Common Bernstein Functions.- 5 Linear Viscoelasticity.- 5.1 Balance of Momentum and Constitutive Laws.- 5.2 Material Functions.- 5.3 Energy Balance and Thermoviscoelasticity.- 5.4 Some One-dimensional Problems.- 5.5 Heat Conduction in Materials with Memory.- 5.6 Synchronous and Incompressible Materials.- 5.7 A Simple Control Problem.- 5.8 Comments.- II Nonscalar Equations.- 6 Hyperbolic Equations of Nonscalar Type.- 6.1 Resolvents of Nonscalar Equations.- 6.2 Well-posedness and Variation of Parameters Formulae.- 6.3 Hyperbolic Perturbation Results.- 6.4 The Generation Theorem.- 6.5 Convergence of Resolvents.- 6.6 Kernels of Positive Type in Hilbert spaces.- 6.7 Hyperbolic Problems of Variational Type.- 6.8 Comments.- 7 Nonscalar Parabolic Equations.- 7.1 Analytic Resolvents.- 7.2 Parabolic Equations.- 7.3 Parabolic Problems of Variational Type.- 7.4 Maximal Regularity of Perturbed Parabolic Problems.- 7.5 Resolvents for Perturbed Parabolic Problems.- 7.6 Uniform Bounds for the Resolvent.- 7.7 Comments.- 8 Parabolic Problems in Lp-Spaces.- 8.1 Operators with Bounded Imaginary Powers.- 8.2 Vector-Valued Multiplier Theorems.- 8.3 Sums of Commuting Linear Operators.- 8.4 Volterra Operators in Lp.- 8.5 Maximal Regularity in Lp.- 8.6 Strong Lp-Stability on the Halfline.- 8.7 Comments.- 9 Viscoelasticity and Electrodynamics with Memory.- 9.1 Viscoelastic Beams.- 9.2 Viscoelastic Plates.- 9.3 Thermoviscoelasticity: Strong Approach.- 9.4 Thermoviscoelasticity: Variational Approach.- 9.5 Electrodynamics with Memory.- 9.6 A Transmission Problem for Media with Memory.- 9.7 Comments.- III Equations on the Line.- 10 Integrability of Resolvents.- 10.1 Stability on the Halfline.- 10.2 Parabolic Equations of Scalar Type.- 10.3 Subordinated Resolvents.- 10.4 Strong Integrability in Hilbert Spaces.- 10.5 Nonscalar Parabolic Problems.- 10.6 Comments.- 11 Limiting Equations.- 11.1 Homogeneous Spaces.- 11.2 Admissibility.- 11.3 A-Kernels for Compact A.- 11.4 Almost Periodic Solutions.- 11.5 Nonresonant Problems.- 11.6 Asymptotic Equivalence.- 11.7 Comments.- 12 Admissibility of Function Spaces.- 12.1 Perturbations: Hyperbolic Case.- 12.2 Subordinated Equations.- 12.3 Admissibility in Hilbert Spaces.- 12.4 A-kernels for Parabolic Problems.- 12.5 Maximal Regularity on the Line.- 12.6 Perturbations: Parabolic Case.- 12.7 Comments.- 13 Further Applications and Complements.- 13.1 Viscoelastic Timoshenko Beams.- 13.2 Heat Conduction in Materials with Memory.- 13.3 Electrodynamics with Memory.- 13.4 Ergodic Theory.- 13.5 Semilinear Equations.- 13.6 Semigroup Approaches.- 13.7 Nonlinear Equations with Accretive Operators.
Présentation de l'éditeur:
This book deals with evolutionary systems whose equation of state can be formulated as a linear Volterra equation in a Banach space. The main feature of the kernels involved is that they consist of unbounded linear operators. The aim is a coherent presentation of the state of art of the theory including detailed proofs and its applications to problems from mathematical physics, such as viscoelasticity, heat conduction, and electrodynamics with memory. The importance of evolutionary integral equations? which form a larger class than do evolution equations? stems from such applications and therefore special emphasis is placed on these. A number of models are derived and, by means of the developed theory, discussed thoroughly. An annotated bibliography containing 450 entries increases the book's value as an incisive reference text. - This excellent book presents a general approach to linear evolutionary systems, with an emphasis on infinite-dimensional systems with time delays, such as those occurring in linear viscoelasticity with or without thermal effects. It gives a very natural and mature extension of the usual semigroup approach to a more general class of infinite-dimensional evolutionary systems. This is the first appearance in the form of a monograph of this recently developed theory. A substantial part of the results are due to the author, or are even new. ( ? ) It is not a book that one reads in a few days. Rather, it should be considered as an investment with lasting value. (Zentralblatt MATH) In this book, the author, who has been at the forefront of research on these problems for the last decade, has collected, and in many places extended, the known theory for these equations. In addition, he has provided a framework that allows one to relate and evaluate diverse results in the literature. (Mathematical Reviews) This book constitutes a highly valuable addition to the existing literature on the theory of
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