Long-Time Dynamics For A Class Of Extensible Beam Equations With Damping

The paper investigates the long-time dynamics of the extensible beam equations with nonlinear structural damping:(?)where к ∈[0,1],θ ∈[1,2),Ω is a bounded domain in RN with the smooth boundary(?)Ω.f(u)is a nonlinear term,and ∣f1(u)∣ ≤ c(1+∣u∣ρ-1),h(x)is an external force term.When θ∈[1,2),1≤p＜pθ=N+4θ/(N-4θ)+,we prove the well-posedness of problem(0.1).We prove that there exists a family of global finite-dimensional attractors in the natural energy space.Morever,the upper semicontinuities of global attractors are proved on the dispersion parameter θ∈[1,2),and perturbation parameter κ∈[0,1],separately.When 1≤p＜p1θ=N-2(θ+1)/(N-2(θ+1)+,we establish a family of exponential attractors and show their stability on the perturbation parameter κ.Besides,when θ= 1,1<p<p*andθ∈(1,2),1≤p≤p*=N+4/N-4,we establish the existence of the global attractors and the exponential attractors in Hθ = V2+θ×Vθ.Finally,we prove the stability of the global attractors and the exponential attractors on the perturbation parameter κ∈[0,1].