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

¹ Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, P.R. China
bzguo@iss.ac.cn
² School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa
³ School of Mathematical Science, Heilongjiang University, Harbin 150080, P.R. China
⁴ School of Mathematical Sciences, Shanxi University, Taiyuan 030006, P.R. China

Received: 30 August 2010

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.