This monograph evolved over the past five years. It had its origin as a set of lecture notes prepared for the Ninth Summer School of Mathematical Physics held at Ravello, Italy, in 1984 and was further refined in seminars and lectures given primarily at the University of Colorado. The material presented is the product of a single mathematical question raised by Dave Kassoy over ten years ago. This question and its partial resolution led to a successful, exciting, almost unique interdisciplinary col laborative scientific effort. The mathematical models described are often times deceptively simple in appearance. But they exhibit a mathematical richness and beauty that belies that simplicity and affirms their physical significance. The mathe matical tools required to resolve the various problems raised are diverse, and no systematic attempt is made to give the necessary mathematical background. The unifying theme of the monograph is the set of models themselves. This monograph would never have come to fruition without the enthu siasm and drive of Dave Eberly-a former student, now collaborator and coauthor-and without several significant breakthroughs in our understand ing of the phenomena of blowup or thermal runaway which certain models discussed possess. A collaborator and former student who has made significant contribu tions throughout is Alberto Bressan. There are many other collaborators William Troy, Watson Fulks, Andrew Lacey, Klaus Schmitt-and former students-Paul Talaga and Richard Ely-who must be acknowledged and thanked.

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This book systematically develops models of Spatially-varying transient processes describing thermal events. Such events should be entirely predictable for a given set of physical properties, system geometry, and initial-boundary conditions. For the various initial-boundary value problems which model a reactive thermal event, the following questions are addressed: 1. Do the models give a reasonable time-history description of the state of the system? 2. Does a particular model distinguish between explosive and nonexplosive thermal events? 3. If the thermal event is explosive, can one predict where the explosion will occur, determine where the hotspots will develop, and finally predict how the hotspot of blowup singularities evolve? Primary emphasis is placed on explosive thermal events and we refer to the three aspects of such events as Blowup - When, Where, and How.

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