This paper settles a conjecture by Gazzola and Pavani  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.
Radu, P., Toundykov, D., & Trageser, J. (2014). Finite time blow-up in nonlinear suspension bridge models. Journal of Differential Equations, 257(11), 4030–4063. https://doi.org/10.1016/j.jde.2014.07.017