Abstract
This paper is dedicated to the investigation of the asymptotic behavior of a delayed wave equation without the presence of any displacement term. First, it is shown that the problem is well-posed in the sense of semigroups theory. Thereafter, LaSalle’s invariance principle is invoked in order to establish the asymptotic convergence for the solutions of the system to a stationary position which depends on the initial data. More importantly, without any geometric condition such as BLR condition (Bardos et al. in SIAM J Control Optim 30:1024–1064, 1992; Lebeau and Robbiano in Duke Math J 86:465–491, 1997) in the control zone, the logarithmic convergence is proved by using an interpolation inequality combined with a resolvent method.
Original language | English |
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Article number | 117 |
Journal | Zeitschrift fur Angewandte Mathematik und Physik |
Volume | 68 |
Issue number | 5 |
DOIs | |
Publication status | Published - Oct 1 2017 |
Externally published | Yes |
Keywords
- Asymptotic behavior
- Logarithmic stability
- Time-delay
- Wave equation
ASJC Scopus subject areas
- General Mathematics
- General Physics and Astronomy
- Applied Mathematics