Singular dynamics of strongly damped beam equation

The paper is devoted to the dynamics of the model for a beam with strong dampingequation(Pε)ε2utt+εδut+αuxxxx+utxxxx−[g(∫0luξ2dξ)+εσ∫0lutξuξdξ]uxx=0,(x,t)∈]0,l[×]0,∞[, where g:R→Rg:R→R is continuously differentiable, δ,σ∈Rδ,σ∈R and α,l,ε>0α,l,ε>0, subject to boundary conditions corresponding to hinged or clamped ends.We show that for ε→0+ε→0+ the dynamics of the equation is close to the dynamics of equationequation(P0)ut=−αu−g(∫0luξ2dξ)A−1/2u, where Au:=uxxxxAu:=uxxxx with the domain determined by one of the above boundary conditions. Specifically, we show that isolated invariant sets of (P0) continue to isolated invariant sets of (Pε), ε>0ε>0 small, having the same Conley index. Moreover, isolated Morse decompositions with respect to (P0) continue to isolated Morse decompositions of (Pε), ε>0ε>0 small, having isomorphic homology index braids.Under some additional assumptions we establish existence and upper semicontinuity results for attractors of  and , ε>0ε>0 small, extending previous results by Ševčovič.