Asymptotic behavior for a past history viscoelastic problem with acoustic boundary conditions

ABSTRACT

In this paper, we consider a viscoelastic equation of variable coefficients in the presence of infinite memory (past history) with nonlinear damping term and nonlinear delay term in the boundary feedback and acoustic boundary conditions. Under suitable assumptions, two arbitrary decay results of the energy solution are established via suitable Lyapunov functionals and some properties of the convex functions. The first stability result is given with a relation between the damping term and relaxation function and the second result is given without imposing any restrictive growth assumption on the damping term and the kernel function g. Our result extends the decay result obtained for problems with finite history to those with infinite history.

KEYWORDS:

• Acoustic boundary conditions
• general decay
• delay term
• past history
• variable coefficients

AMS SUBJECT CLASSIFICATIONS:

• 35L70
• 35B40
• 93D15

Acknowledgements

The authors are highly grateful to the anonymous referee for his/her valuable comments and suggestions for the improvement of the paper.