Historically, there have been three major strategies for dealing with Cauchy problems that are not well-posed in file classical sense: semigroup, abstract distribution, and regularization methods. Semigroup and distribution methods tenture 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 in semigroup, distribution, and ill-posed problems theories, 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, 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.