ABSTRACT
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 10.1 Introducing (A-HCP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 10.2 Defining eAt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 10.3 Properties of eAt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 10.4 The Abstract Homogeneous Cauchy Problem: Well-posedness . . . . . . . . . . . . . . . . 294 10.5 A Brief Glimpse of Long-Term Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 10.6 Looking Ahead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
We have encountered a large collection of IBVPs arising in the mathematical modeling of very different phenomena that can be all be reformulated as a single abstract evolution equation of the form (A-HCP), or a closely-related variant. Such commonality prompts us to determine if a theory analogous to Part I could be formulated to study all of these problems under the same theoretical umbrella. At the end of Chapter 9, we made some preliminary observations that gave us hope that such a theory could be developed. We make these observations more formal in this chapter.
