ABSTRACT

Using a modest dose of linear semigroup theory, we were able to develop a rather rich existence theory that formed a theoretical basis for a formal mathematical study of vastly different phenomena. We introduced a significant amount of complexity into the IBVPs by way of time delays, perturbations, and complex forcing terms. But, as rich as the theory is, it all hinges on the crucial assumption that the operator A : dom(A) ⊂X →X is linear and generates a C0−semigroup on X . The problem is that these two assumptions do not hold for many phenomena, including the various improvements on the models discussed throughout the text. As such, the question is whether or not we can somehow argue analogously as we did when generalizing the setting of Chapter 2 to Chapter 3 to develop a theory in the so-called nonlinear case. The answer is a tentative yes, but the extension from the linear to the nonlinear setting takes place on a much grander scale than the generalization of the finite-dimensional to the infinite-dimensional setting in the linear case and requires a considerably higher degree of sophistication. We shall explore how this theory unfolds for some basic nonlinear models in this chapter.