Spatial behavior for a fourth-order dispersive equation

In this note we investigate the spatial behavior of a linear equation of fourth order which models several mechanical situations when dispersive and dissipative effects are taken into account. In particular, this equation models the extensional vibration of a bar when we assume that external friction, with a rough substrate for example, is present. We show that for such an equation a Phragmen-Lindelof alternative of exponential type can be obtained. A bound for the amplitude term in terms of boundary data is obtained. Moreover, when friction is absent, we obtain exponential decay results in the case of harmonic vibrations and we prove a polynomial decay estimate for general solutions.