Existence Families, Functional Calculi and Evolution EquationsThis 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. |
Contents
Intuition and Elementary Examples | 1 |
Existence Families | 7 |
Regularized Semigroups | 13 |
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a-type abstract Cauchy problem Algebraic Banach space bounded operators bounded strongly continuous C₁ Cauchy problem 0.1 closed operator Co(R commutes construction contained continuous holomorphic semigroup continuously embedded converges Corollary dense set dense solution set densely defined equicontinuous equipartition of energy Example exponentially bounded holomorphic family of operators following are equivalent fractional powers Frechet space functional calculus heat equation Im(C implies initial data injective Laplace transform left half-plane Lemma Lipschitz continuous Math matrix mild C-existence family n-times integrated semigroup nonempty norm Note numerical range pc(A Proceedings Proof of Theorem Proposition 3.10 Re(z regularized semigroup Schrödinger semigroup of angle semigroup see Chapter solution of 0.1 spectral spectrum strongly continuous group strongly continuous holomorphic strongly continuous semigroup Suppose Theorem 3.4 topology unbounded uniformly continuous unique solution W₁(t well-posed well-posedness მო