This monograph, "Evolutionary Integral Equations and Applications," by Jan Prüss, is a comprehensive treatment of abstract Volterra integral equations and their applications in mathematical physics. The book is divided into three main chapters: Equations of Scalar Type, Nonscalar Problems, and Equations on the Line. 1. **Equations of Scalar Type**: This chapter focuses on equations of the form \( u(t) = \int_0^t A(\tau) u(t - \tau) d\tau + f(t) \) on the halfline and \( v(t) = \int_0^\infty A(\tau) v(t - \tau) d\tau + g(t) \) on the line. It covers well-posedness, resolvents, analytic resolvents, parabolic equations, subordination, and viscoelasticity. Key results include the existence and properties of resolvents, the variation of parameters formula, and the subordination principle. 2. **Nonscalar Problems**: This chapter addresses more complex problems with operator-valued kernels \( A \in L_{loc}^1(\mathbb{R}_+; \mathcal{B}(Y, X)) \). It introduces strong and mild solutions, well-posedness, resolvents, and pseudo-resolvents. The chapter also discusses energy inequalities, coercivity, and perturbation results. Applications include viscoelasticity, thermoviscoelasticity, and electrodynamics with memory. 3. **Equations on the Line**: This chapter explores the equation \( v(t) = \int_0^\infty A(\tau) v(t - \tau) d\tau + g(t) \) on the line and its relationship to the halfline equation. It covers integrability of resolvents, limiting equations, and stability properties. Key results include necessary conditions for integrability, the connection between resolvents and Laplace transforms, and the asymptotic behavior of solutions. The book provides a detailed theoretical framework and numerous applications, making it a valuable resource for researchers and students in the fields of mathematical physics, functional analysis, and applied mathematics.This monograph, "Evolutionary Integral Equations and Applications," by Jan Prüss, is a comprehensive treatment of abstract Volterra integral equations and their applications in mathematical physics. The book is divided into three main chapters: Equations of Scalar Type, Nonscalar Problems, and Equations on the Line. 1. **Equations of Scalar Type**: This chapter focuses on equations of the form \( u(t) = \int_0^t A(\tau) u(t - \tau) d\tau + f(t) \) on the halfline and \( v(t) = \int_0^\infty A(\tau) v(t - \tau) d\tau + g(t) \) on the line. It covers well-posedness, resolvents, analytic resolvents, parabolic equations, subordination, and viscoelasticity. Key results include the existence and properties of resolvents, the variation of parameters formula, and the subordination principle. 2. **Nonscalar Problems**: This chapter addresses more complex problems with operator-valued kernels \( A \in L_{loc}^1(\mathbb{R}_+; \mathcal{B}(Y, X)) \). It introduces strong and mild solutions, well-posedness, resolvents, and pseudo-resolvents. The chapter also discusses energy inequalities, coercivity, and perturbation results. Applications include viscoelasticity, thermoviscoelasticity, and electrodynamics with memory. 3. **Equations on the Line**: This chapter explores the equation \( v(t) = \int_0^\infty A(\tau) v(t - \tau) d\tau + g(t) \) on the line and its relationship to the halfline equation. It covers integrability of resolvents, limiting equations, and stability properties. Key results include necessary conditions for integrability, the connection between resolvents and Laplace transforms, and the asymptotic behavior of solutions. The book provides a detailed theoretical framework and numerous applications, making it a valuable resource for researchers and students in the fields of mathematical physics, functional analysis, and applied mathematics.