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 ? ?[0,1],? ?[1,2),? is a bounded domain in RN with the smooth boundary(?)?.f(u)is a nonlinear term,and ?f1(u)? ? c(1+?u??-1),h(x)is an external force term.When ??[1,2),1?p?p?=N+4?/(N-4?)+,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 ??[1,2),and perturbation parameter ??[0,1],separately.When 1?p?p1?=N-2(?+1)/(N-2(?+1)+,we establish a family of exponential attractors and show their stability on the perturbation parameter ?.Besides,when ?= 1,1<p<p*and??(1,2),1?p?p*=N+4/N-4,we establish the existence of the global attractors and the exponential attractors in H? = V2+?×V?.Finally,we prove the stability of the global attractors and the exponential attractors on the perturbation parameter ??[0,1].