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Spatial behavior for a fourth-order dispersive equation. (English) Zbl 1111.35091

Summary: 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 Phragmén-Lindelöf 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.

MSC:

35Q72 Other PDE from mechanics (MSC2000)
74J05 Linear waves in solid mechanics
74H45 Vibrations in dynamical problems in solid mechanics
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