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Relevant to a variety of mathematical models in physics, engineering, and finance, this volume studies Cauchy problems that are not well-posed in the classical sense. It brings together and examines three major approaches to treating such problems: semigroup methods, abstract distribution methods, and regularization methods. Although extensively developed over the last decade, the authors provide a unique, self-contained account of these methods and demonstrate the profound connections between them. Accessible to beginning graduate students, this volume brings together many different ideas to serve as a reference on modern methods for abstract linear evolution equations.