Singular dynamics of strongly damped beam equation

The paper is devoted to the dynamics of the model for a beam with strong damping epsilon(2)u(tt) + epsilon delta u(t) + alpha u(xxxx) + u(txxxx) - [g(integral(t)(0) u(xi)(2) d xi) + epsilon sigma integral(t)(0) u(t xi) u(xi) d(xi)]u(xx) = 0. (x. t) is an element of ]0, l[ x ]0,infinity[, (P-epsilon) where g : R -> R is continuously differentiable, delta, sigma is an element of R and alpha, l, epsilon > 0. subject to boundary conditions corresponding to hinged or clamped ends. We show that for epsilon -> 0(+) the dynamics of the equation is close to the dynamics of equation u(t) = -alpha u -g(integral(t)(0) mu(2)(xi) d xi) A(-1/2) u, (P-0) where Au := u(xxxx) with the domain determined by one of the above boundary conditions. Specifically, we show that isolated invariant sets of (P-0) continue to isolated invariant sets of (P-epsilon), epsilon > 0 small, having the same Conley index. Moreover, isolated Morse decompositions with respect to (P-0) continue to isolated Morse decompositions of (P-epsilon) epsilon > 0 small, having isomorphic homology index braids. Under some additional assumptions we establish existence and upper semicontinuity results for attractors of (P-0) and (P-epsilon), epsilon > 0 small, extending previous results by Sevcovic. (C) 2009 Elsevier Inc. All rights reserved.