Indirect stabilization of a coupled system by memory effects

Abstract

We consider an abstract model of two coupled elastic materials. One of the materials has conservative characteristics, whereas the other one has dissipative properties. The dissipative effect is caused by the presence of a memory term that depends on the fractional stationary operator with exponent $\theta \in [0,1]$. In this paper, we study the asymptotic behavior of the solutions for this system. We show that the solutions decay polynomially with the rate $t^{-1/(4-2\theta )}$. For problems with that level of generality, we show that the above rate is the best. We also study the asymptotic behavior when the wave propagation speeds of both materials coincide. For this case, we find that the decay rate is so fast as $t^{-1/(2-2\theta )}$ for $\theta \ne 1$. For completeness, we also approach the case $\theta =1$, where an exponential decay of solutions is obtained.

Keywords:

• Indirect stabilization
• coupled equations
• memory term
• asymptotic stability
• decay rates

Mathematics Subject Classifications:

• 35B40
• 35L90
• 35L53
• 35Q93

Acknowledgments

We thank the reviewer for his/her review and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), Brazil - Finance Code 001. The second author was partially support by Conselho Nacional de Desenvolvimento Cientifico e Tecnológico (CNPq), Brazil [grant number 304883/2019-0].