Finite time blow-up in nonlinear suspension bridge models

Abstract

This paper settles a conjecture by Gazzola and Pavani [10] regarding solutions to the fourth order ODE $w^{(4)}+k{w}^{″}+f(w)=0$ 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 $k\le 0$ 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.