Finite time blow-up in nonlinear suspension bridge models

This paper settles a conjecture by Gazzola and Pavani [10] regarding solutions to the fourth order ODE w(4) + kw′′ + f (w) = 0 which arises in models of traveling waves in suspension bridges when k > 0. Under suitable assumptions on the nonlinearity f and initial data, we demonstrate blow-up in finite time. The case k ≤ 0 was first investigated by Gazzola et al., and it is also handled here with a proof that requires less differentiability on f . Our approach is inspired by Gazzola et al. and exhibits the oscillatory mechanism underlying the finite-time blow-up. This blow-up is nonmonotone, with solutions oscillating to higher amplitudes over shrinking time intervals. In the context of bridge dynamics this phenomenon appears to be a consequence of mutually-amplifying interactions between vertical displacements and torsional oscillations. © 2014 Elsevier Inc. All rights reserved. MSC: 34A12; 35C07; 34C10