This work presents a few facts on that theory and some applications. No claim of completeness is made, either in the text or in the references; many important results have been left out and many important papers are not mentioned. Chapter I expounds semigroup theory; Chapter II presents cosine function theory, which stands in relation to the second order equation as semigroup theory stands in relation to the first order equation. Chapter III deals with the reduction of to a first order system mentioned above and other related topics. The next four chapters are on applications; in Chapter IV we treat the initial-boundary value problem with A a second order uniformly elliptic partial differential operator in a domain of m-dimensional Euclidean space, with either the Dirichlet boundary condition or a variational boundary condition. Chapter V treats the second order equation in Hilbert spaces, where many special results are available; there are applications to equations with almost periodic and periodic solutions. Chapters VI and VII are on singular perturbation problems, with applications to diverse physical situations. Finally, in Chapter VIII we touch upon the theory of the "ctmplete" second order equation without going too far into it; mostly, we search for the correct definition of correctly posed initial value problem. Some shortcuts through the book are possible, and we do not bother to indicate them explicitly; for instance, Chapter III is only briefly needed in Chapters IV and V and not used at all in Chapters VI and VII.
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