Stability for a Class of Extensible Beams with Degenerate Nonlocal Damping

This paper investigates the well-posedness and stability of the beam model with degenerate nonlocal damping: u tt + Δ 2 u - M ( ‖ ∇ u ‖ 2 ) Δ u + ( ‖ Δ u ‖ θ + q ‖ u t ‖ ρ ) ( - Δ ) δ u t + f ( u ) = 0 in Ω × R + , where Ω ⊂ R n is a bounded domain with smooth boundary, θ ≥ 1 , q ≥ 0 , ρ > 0 and 0 ≤ δ ≤ 1 . The main purpose in the present paper is to show that the transition from the case q = 0 to the case q > 0 produces an explicit influence on the stability of energy solutions. More precisely, when q = 0 , we conclude that the energy goes to zero as t goes to infinity without an explicit decay rate; while when q > 0 , we present a polynomial decay rate of type ( 1 + t ) - 2 ρ that depends only on the exponent ρ of the velocity term, not on θ and δ . Furthermore, we prove that the energy cannot be exponentially stable and derive more accurate decay rates of the energy.