TY - JOUR

T1 - On the exponential and polynomial convergence for a delayed wave equation without displacement

AU - Ammari, Kaïs

AU - Chentouf, Boumediène

N1 - Funding Information:
This work was supported and funded by Kuwait University , Research Project No. ( SM04/17 ).
Publisher Copyright:
© 2018 Elsevier Ltd

PY - 2018/12

Y1 - 2018/12

N2 - This article places primary emphasis on improving the asymptotic behavior of a multi-dimensional delayed wave equation in the absence of any displacement term. In the first instance, the delay is assumed to occur in the boundary. Then, invoking Bardos–Lebeau–Rauch (BLR) geometric condition (Bardos et al., 1992; Lebeau and Robbiano, 1997) on the domain, the exponential convergence of solutions to their equilibrium state is proved. In turn, an internal delayed wave equation is considered in the second instance, where the three-dimensional domain possesses trapped ray and hence the (BLR) geometric condition (Bardos et al., 1992, Lebeau and Robbiano, 1997) does not hold. Moreover, the internal damping is localized. In such a situation, polynomial convergence results are established. These two findings improve earlier results of Ammari and Chentouf (2017), Phung (2017) and Stahn (2017).

AB - This article places primary emphasis on improving the asymptotic behavior of a multi-dimensional delayed wave equation in the absence of any displacement term. In the first instance, the delay is assumed to occur in the boundary. Then, invoking Bardos–Lebeau–Rauch (BLR) geometric condition (Bardos et al., 1992; Lebeau and Robbiano, 1997) on the domain, the exponential convergence of solutions to their equilibrium state is proved. In turn, an internal delayed wave equation is considered in the second instance, where the three-dimensional domain possesses trapped ray and hence the (BLR) geometric condition (Bardos et al., 1992, Lebeau and Robbiano, 1997) does not hold. Moreover, the internal damping is localized. In such a situation, polynomial convergence results are established. These two findings improve earlier results of Ammari and Chentouf (2017), Phung (2017) and Stahn (2017).

KW - Asymptotic behavior

KW - Exponential convergence

KW - Polynomial convergence

KW - Time-delay

KW - Wave equation

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U2 - 10.1016/j.aml.2018.06.021

DO - 10.1016/j.aml.2018.06.021

M3 - Article

AN - SCOPUS:85049481007

SN - 0893-9659

VL - 86

SP - 126

EP - 133

JO - Applied Mathematics Letters

JF - Applied Mathematics Letters

ER -