Open Access
Translator Disclaimer
2000 Riesz basis property of root vectors of nonselfadjoint operators generated by radial damped wave equations
Marianna A. Shubov
Adv. Differential Equations 5(4-6): 623-656 (2000). DOI: 10.57262/ade/1356651342

Abstract

We consider an infinite sequence of radial wave equations obtained by the separation of variables in the spherical coordinates from the 3-dimensional damped wave equation with spatially nonhomogeneous spherically symmetric coefficients. Our main objects of interest are the nonselfadjoint operators in the energy spaces of 2-component initial data, which are the dynamics generators for the systems governed by the aforementioned equations and nonselfadjoint boundary conditions on the sphere $r=a$. Our main result is the fact that the sets of the root vectors (generalized eigenvectors) of these operators form Riesz bases in the corresponding energy spaces. This result has several applications. The first one is the fact that the aforementioned operators are spectral in the sense of N. Dunford, and, therefore, we have a new nontrivial class of spectral operators. Another application is a precise estimate on the rate of the energy decay, which is equal to the spectral abscissa of the corresponding semigroup. Finally, we use the results of our spectral analysis to formulate the solutions of several problems in control theory of systems governed by damped wave equations (the proofs are given in another work).

Citation

Download Citation

Marianna A. Shubov. "Riesz basis property of root vectors of nonselfadjoint operators generated by radial damped wave equations." Adv. Differential Equations 5 (4-6) 623 - 656, 2000. https://doi.org/10.57262/ade/1356651342

Information

Published: 2000
First available in Project Euclid: 27 December 2012

zbMATH: 0996.47043
MathSciNet: MR1750113
Digital Object Identifier: 10.57262/ade/1356651342

Subjects:
Primary: 47F05
Secondary: 35P99 , 47B25 , 47B40

Rights: Copyright © 2000 Khayyam Publishing, Inc.

JOURNAL ARTICLE
34 PAGES


SHARE
Vol.5 • No. 4-6 • 2000
Back to Top