Intrinsic decay rates for the energy of a nonlinear viscoelastic equation modeling the vibrations of thin rods with variable density

We consider the long-time behavior of a nonlinear PDE with a memory term which can be recast in the abstract form d d ⁢ t ⁢ ρ ⁢ ( u t ) + A ⁢ u t ⁢ t + γ ⁢ A θ ⁢ u t + A ⁢ u - ∫ 0 t g ⁢ ( s ) ⁢ A ⁢ u ⁢ ( t - s ) = 0 , d/dtρ(uₜ)+Auₜₜ+γ Aθuₜ+Au-∫₀ᵗg(s)Au(t% -s)=0, where A is a self-adjoint, positive definite operator acting on a Hilbert space H , ρ ⁢ ( s ) ρ(s) is a continuous, monotone increasing function, and the relaxation kernel g ⁢ ( s ) g(s) is a continuous, decreasing function in L 1 ⁢ ( ℝ + ) L₁(R₊) with g ⁢ ( 0 ) > 0 g(0)>0 . Of particular interest is the case when A = - Δ