Existence Families, Functional Calculi and Evolution Equations [electronic resource] / by Ralph deLaubenfels.

Intuition and elementary examples -- Existence families -- Regularized semigroups -- The solution space of an operator and automatic well-posedness -- Exponentially bounded (Banach) solution spaces -- Well-posedness on a larger space; Generalized solutions -- Entire vectors and entire existence families -- Reversibility of parabolic problems -- The cauchy problem for the Laplace equation -- Boundary values of holomorphic semigroups -- The Schrödinger equation -- Functional calculus for commuting generators of bounded strongly continuous groups -- Petrovsky correct matrices of generators of bounded strongly continuous groups -- Arbitrary matrices of generators of bounded strongly continuous groups -- More examples of regularized semigroups -- Existence and uniqueness families -- C-resolvents and Hille-Yosida type theorems -- Relationship to integrated semigroups -- Perturbations -- Type of an operator -- Holomorphic C-existence families -- Unbounded holomorphic functional calculus for operators with polynomially bounded resolvents -- Spectral conditions guaranteeing solutions of the abstract Cauchy problem -- Polynomials of generators -- Iterated abstract Cauchy problems -- Equipartition of energy -- Simultaneous solution space -- Exponentially bounded simultaneous solution space -- Simultaneous existence families -- Simultaneous existence families for matrices of operators -- Time dependent evolution equations.

This book presents an operator-theoretic approach to ill-posed evolution equations. It presents the basic theory, and the more surprising examples, of generalizations of strongly continuous semigroups known as 'existent families' and 'regularized semigroups'. These families of operators may be used either to produce all initial data for which a solution in the original space exists, or to construct a maximal subspace on which the problem is well-posed. Regularized semigroups are also used to construct functional, or operational, calculi for unbounded operators. The book takes an intuitive and constructive approach by emphasizing the interaction between functional calculus constructions and evolution equations. One thinks of a semigroup generated by A as etA and thinks of a regularized semigroup generated by A as etA g(A), producing solutions of the abstract Cauchy problem for initial data in the image of g(A). Material that is scattered throughout numerous papers is brought together and presented in a fresh, organized way, together with a great deal of new material.