On linear problems with surface dissipation of energy

O. A. Andronova ; N. D. Kopachevskii

Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 29 (2008), pp. 11-28

Résumé

The first part of this work is devoted to applications of functional analysis methods to a linear initial-boundary value problem of mathematical physics with a surface dissipation of the energy. Its abstract analog is studied as well. The abstract Green formula for a triple of Hilbert spaces is used. In the second part, spectral problems generated by linear initial-boundary value problems with a surface dissipation of the energy are studied. First we formulate the spectral problem of mathematical physics and the corresponding abstract problem. Further, we consider basic properties of the spectrum and show that it is rather specific in the case of considered problems; particular examples (one-dimensional and two-dimensional ones as well as an example of a cylindrical domain) are used for that. It turns out that the spectrum migrates in the complex plane, while the dissipation parameter changes from zero to infinity. Examples of numerical computations of the spectrum by means of the iteration method are provided. Further, we investigate the general setting of the spectral problem. Using a general result of Azizov, we prove that the spectrum of the generic problem is discrete and has a limiting point at infinity.