Attractors and their properties for a class of nonlocal extensible beams

This paper is concerned with well-posedness and asymptoticbehavior to a class of extensible beams with nonlinear fractionaldamping and source termsuₜₜ + Δ² u - M(∫Ω|∇ u|² \, dx)Δ u + N(∫Ω|∇ u|² \, dx )(-Δ)θ uₜ + f(u) = hin Ω×(0,∞), where Ω⊂ Rq isa bounded domain with smooth boundary ∂Ω, 0≤θ ≤ 1, M and N are scalar functions, $f(u)$ is asource term and h is a forcing term. When θ=1 we havethe classical strong damping -Δ uₜ, and if θ=0 thenweak damping uₜ is considered. Under the mathematical point ofview the abstract setting of (-Δ)θuₜwith 0< θ < 1 constitutes a way to change thecharacteristic of the problem in wave and plate vibrations, andmaybe the decay rate of solutions mainly when we considernonconstant coefficient damping N. The main purpose in thepresent paper is to show that the transition from the case 0<θ ≤ 1 to the case θ=0 does not produce anyinfluence on the asymptotic behavior of solutions, that is, allresults are obtained by moving θ∈[0,1] uniformly. Ourmain result establishes the existence of finite-dimensionalcompact global and exponential attractors by using an approach onquasi-stable dynamical systems given by Chueshov and Lasiecka(2010).