General and optimal decay for a quasilinear parabolic viscoelastic system

Abderrahmane Youkana; Salim A. Messaoudi

doi:10.3934/dcdss.2021129

Article Contents

2022, Volume 15, Issue 5: 1307-1316. Doi: 10.3934/dcdss.2021129

This issue Previous Article Boundary observability and exact controllability of strongly coupled wave equations Next Article

General and optimal decay for a quasilinear parabolic viscoelastic system

Abderrahmane Youkana^1, , and
Salim A. Messaoudi^2,

1.
Department of Engineering Processes, University of Bejaia, Bejaia 06000, Algeria
2.
Department of Mathematics, University of Sharjah, Sharjah, P.O. Box 27272, UAE

^* Corresponding author: Abderrahmane Youkana
^* Corresponding author: Abderrahmane Youkana

Received: May 31, 2021

Revised: July 31, 2021

Early access: October 2021

Published: May 2022

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Abstract

In this paper, we give a general decay rate for a quasilinear parabolic viscoelatic system under a general assumption on the relaxation functions satisfying $ g'(t) \leq - \xi(t) H(g(t)) $, where $ H $ is an increasing, convex function and $ \xi $ is a nonincreasing function. Precisely, we establish a general and optimal decay result for a large class of relaxation functions which improves and generalizes several stability results in the literature. In particular, our result extends an earlier one in the literature, namely, the case of the polynomial rates when $ H(t) = t^p, \ t\geq 0, \forall p>1 $, instead the parameter $ p \in [1, \frac{3}{2}[ $.

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Mathematics Subject Classification: 35K05.

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References

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