Although the theory of well-posed Cauchy problems is reasonably understood, ill-posed problems-involved in a numerous mathematical models in physics, engineering, and finance- can be approached in a variety of ways. Historically, there have been three major strategies for dealing with such problems: semigroup, abstract distribution, and regularization methods. Semigroup and distribution methods restore well-posedness, in a modern weak sense. Regularization methods provide approximate solutions to ill-posed problems. Although these approaches were extensively developed over the last decades by many researchers, nowhere could one find a comprehensive treatment of all three approaches.
Abstract Cauchy Problems: Three Approaches provides an innovative, self-contained account of these methods and, furthermore, demonstrates and studies some of the profound connections between them. The authors discuss the application of different methods not only to the Cauchy problem that is not well-posed in the classical sense, but also to important generalizations: the Cauchy problem for inclusion and the Cauchy problem for second order equations.
Accessible to nonspecialists and beginning graduate students, this volume brings together many different ideas to serve as a reference on modern methods for abstract linear evolution equations.
Preface. Introduction. ILLUSTRATION AND MOTIVATION. Heat Equation. The Reversed Cauchy Problem for the Heat Equation. Wave Equation. SEMIGROUP METHODS. C0-Semigroups. Integrated Semigroups. k-Convoluted Semigroups. C-Regularized Semigroups. Degenerate Semigroups. The Cauchy Problem for Inclusions. Second Order Equations. ABSTRACT DISTRIBUTION METHODS. The Cauchy Problem. The Degenerate Cauchy Problem. Ultradistributions and New Distributions. REGULARIZATION METHODS. The Ill-Posed Cauchy Problem. Regularization and C-Regularized Semigroups. BIBLIOGRAPY. GLOSSARY OF NOTATION. INDEX.