During the last two decades the theory of abstract Volterra equations has under­ gone rapid development. To a large extent this was due to the applications of this theory to problems in mathematical physics, such as viscoelasticity, heat conduc­ tion in materials with memory, electrodynamics with memory, and to the need of tools to tackle the problems arising in these fields. Many interesting phenomena not found with differential equations but observed in specific examples of Volterra type stimulated research and improved our understanding and knowledge. Al­ though this process is still going on, in particular concerning nonlinear problems, the linear theory has reached a state of maturity. In recent years several good books on Volterra equations have appeared. How­ ever, none of them accounts for linear problems in infinite dimensions, and there­ fore this part of the theory has been available only through the - meanwhile enor­ mous - original literature, so far. The present monograph intends to close this gap. Its aim is a coherent exposition of the state of the art in the linear theory. It brings together and unifies most of the relevant results available at present, and should ease the way through the original literature for anyone intending to work on abstract Volterra equations and its applications. And it exhibits many prob­ lems in the linear theory which have not been solved or even not been considered, so far.

Inhalt

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.