Decay Rate for a Viscoelastic Equation with Strong Damping and Acoustic Boundary Conditions

This paper is concerned with a nonlinear viscoelastic equation with strong damping: ( ) ( ) 0 , d 0, t t tt tt t u u u u g t s u x s s u ρ − ∆ − ∆ + − ∆ − ∆ = ∫ . The objective of the present paper is to provide some results on the long-time behavior to this equation with acoustic boundary conditions. By using the assumptions on the relaxation function due to Tatar [1], we show an arbitrary rate of decay with not necessary of an exponential or polynomial one and without the assumption ( ) 0 1 d 2 g s s ∞ < ∫ condition. The result extends and improves some results given in Cavalcanti [2].