Preprint 2012-05
Margareth Alves, Jaime Muñoz-Rivera, Mauricio Sepúlveda, Octavio Vera, María Zegarra-Garay:
The asymptotic behaviour of the linear transmission problem in viscoelasticity
Abstract:
Here we consider the transmission problem with localized Kelvin Voigts viscoelastic damping. Our main result is to show that the corresponding semigroup $e^{{cal A}t}$ is not exponentially stable, but the solution decays polynomially to zero as $1/(1+t)^2,$ when the initial data is taken over the Domain $mathcal{D}(mathcal{A}).$ Moreover we prove that this rate of decay is optimal. Finally using a second order scheme that ensures the decay of energy (Newmark-$eta$ method), we give some numerical examples which demonstrate this polynomial asymptotic behavior.
This preprint gave rise to the following definitive publication(s):
Margareth ALVES, Jaime MUñOZ-RIVERA, Mauricio SEPúLVEDA, Octavio VERA, María ZEGARRA-GARAY: The asymptotic behaviour of the linear transmission problem in viscoelasticity. Journal of Mathematical Analysis and Applications, vol. 399, 2, pp. 472-479, (2013).
Margareth ALVES, Jaime MUñOZ-RIVERA, Mauricio SEPúLVEDA, Octavio VERA, María ZEGARRA-GARAY: The asymptotic behaviour of the linear transmission problem in viscoelasticity. Mathematische Nachrichten, vol. 287, 5-6, pp. 483-497, (2014).