ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - July 2012 - Volume 18, Issue 03  

Research Article

On Spectrum and Riesz basis property for one-dimensional wave equation with Boltzmann damping ∗

Bao-Zhu Guoa1a2a4 and Guo-Dong Zhanga2a3

a1 Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, P.R. China. bzguo@iss.ac.cn

a2 School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa

a3 School of Mathematical Science, Heilongjiang University, Harbin 150080, P.R. China

a4 School of Mathematical Sciences, Shanxi University, Taiyuan 030006, P.R. China

Abstract

In this paper, we study the one-dimensional wave equation with Boltzmann damping. Two different Boltzmann integrals that represent the memory of materials are considered. The spectral properties for both cases are thoroughly analyzed. It is found that when the memory of system is counted from the infinity, the spectrum of system contains a left half complex plane, which is sharp contrast to the most results in elastic vibration systems that the vibrating dynamics can be considered from the vibration frequency point of view. This suggests us to investigate the system with memory counted from the vibrating starting moment. In the latter case, it is shown that the spectrum of system determines completely the dynamic behavior of the vibration: there is a sequence of generalized eigenfunctions of the system, which forms a Riesz basis for the state space. As the consequences, the spectrum-determined growth condition and exponential stability are concluded. The results of this paper expositorily demonstrate the proper modeling the elastic systems with Boltzmann damping.

(Received August 30 2010)

(Online publication September 27 2011)

Key Words:

  • Wave equation;
  • spectrum;
  • Riesz basis;
  • stability;
  • Boltzmann damping

Mathematics Subject Classification:

  • 35L05;
  • 47A10;
  • 47E05;
  • 58K55

Footnotes

∗  This work was supported by the National Natural Science Foundation of China, the National Basic Research Program of China with grant No. 2011CB808002, and the National Research Foundation of South Africa.