Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Spatial behavior for a fourth-order dispersive equation

Authors: Ramon Quintanilla and Giuseppe Saccomandi
Journal: Quart. Appl. Math. 64 (2006), 547-560
MSC (2000): Primary 35Q72; Secondary 74J05
DOI: https://doi.org/10.1090/S0033-569X-06-01025-2
Published electronically: July 5, 2006
MathSciNet review: 2259054
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Abstract: 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 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.

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Ramon Quintanilla
Affiliation: Matemática Aplicada 2 Universidad Politècnica de Catalunya Colón, 11, Terrassa. Barcelona, Spain
Email: Ramon.Quintanilla@upc.edu

Giuseppe Saccomandi
Affiliation: Dipartimento di Ingegneria dell’Innovazione, Sezione Industriale, Università di Lecce, Italy
Email: giuseppe.saccomandi@unile.it

DOI: https://doi.org/10.1090/S0033-569X-06-01025-2
Keywords: Phragm\'en-Lindel\"of alternative, dispersive effects, friction
Received by editor(s): December 14, 2005
Published electronically: July 5, 2006
Additional Notes: This work is supported by the project “Aspectos de Estabilidad en Termomecánica"(BFM2003-00309), by the GNMF of INDAM and PRIN 2003 “Problemi Matematici Nonlineari e Stabilità nei Modelli del Continuo". GS acknowledges the hospitality and support of UPC.
Article copyright: © Copyright 2006 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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