Evolution Equations

Summary

^*Evolution equations are the abstract formulation of dynamic partial differential equations. In this chapter, we examine both semilinear and nonlinear evolution equations. We start by developing the mathematical tools which are necessary in this study. Among them, the notion of evolution triple plays central role and always allows us to use different spaces within the analysis of a single evolution equation. First we consider semilinear evolutions, which we analyze using the semigroup method. Subsequently, we pass to nonlinear evolutions. We consider two such classes. Evolutions of the subdifferential type (which incorporate variational inequalities) and evolution formulated in the framework of evolution triples with operators of monotone type. The Galerkin method is crucial here. We conclude with the study of second-order evolutions.