On a decay rate for nonlinear extensible viscoelastic beams with history setting

Abstract

In this paper we deal with a nonlinear extensible viscoelastic beam model whose memory term is considered in a history setting. The goal is to extend an approach on stability first provided by Guesmia and Messaoudi (J Math Anal Appl. 2014;416:212–228) to a class of viscoelastic beams/plates with nonlinear extensible and source terms. Our stability result contributes in clarifying how the constants appearing in the decay rate depend upon the nonlinearities and the size of initial data. Thus, it also complements some results dealing with this methodology.

Keywords:

• Extensible beams
• viscoelastic models
• well-posedness
• decay rates

AMS Subject Classifications:

• 35B35
• 35B40
• 35L76
• 74D99

Acknowledgements

The authors would like to thank the referees for their valuable comments that aided the authors to improve and clarify a previous version of the present article.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

E. H. Gomes Tavares has been supported by the Brazilian agency CAPES with a Ph.D. scholarship, M. A. Jorge Silva has been supported by the CNPq [grant number 441414/2014-1]; V. Narciso has been partially supported by the FUNDECT [grant number 219/2016].