Evolutionary Integral Equations and Applications (Monographs in Mathematics)

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.