English | 1994 | ISBN: 3540577033 | PDF | pages: 272 | 9,2 mb
This book presents an operator-theoretic approach to ill-posed evolution equations. It presents the basic theory, & the more surprising examples, of generalizations of strongly continuous semigroups known as 'existent families' & '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 & constructive approach by emphasizing the interaction between functional calculus constructions & evolution equations. One thinks of a semigroup generated by A as etA & 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 & presented in a fresh, organized way, together with a great deal of new material.