The Notion of Energy of Solutions: One of the Most Important Quantities

The main issue of this chapter is the notion of energy of solutions, a very effective tool for the treatment of nonstationary or evolution models. We introduce energies for different models and explain conservation of energies, if possible. Sometimes, the choice of a suitable energy seems to be a miracle, sometimes one has different choices. Do we have energy conservation? Do we have blow up of the energy for t →∞? Do we have a decay of the energy for t →∞? These and related questions will be answered in this chapter. Moreover, it is shown how lower order terms may influence the choice and the long-time behavior of a suitable energy. Here we restrict ourselves to mass and different damping terms. Finally, we explain for several models the long-time behavior of local energies for solutions of mixed problems with Dirichlet condition in exterior domains.