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
Full-text PDF Free Access
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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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- Markus Lazar, Gérard A. Maugin, and Elias C. Aifantis, On a theory of nonlocal elasticity of bi-Helmholtz type and some applications, Internat. J. Solids Structures 43 (2006), no. 6, 1404–1421. MR 2200992, DOI https://doi.org/10.1016/j.ijsolstr.2005.04.027
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AS97 K.A. Ames and B. Straughan. Non-Standard and Improperly Posed Problems. Academic Press, San Diego, (1997).
ADS:02 S.N. Antontsev, J.I. Díaz and S. Shmarev. Energy Methods for Free Boundary Problems: Applications to Nonlinear PDEs and Fluid Mechanics. Birkhäuser, Boston (2002).
DS M. Destrade and G. Saccomandi. Waves and vibrations in a solid of second grade, to appear in Proceedings of WASCOM 2005, World Scientific, Singapore (2005).
Eri A. C. Eringen, Theory of thermo-microstretch fluids and bubbly liquids. International Journal of Engineering Science 28, 133-143 (1990).
FKP89 J. N. Flavin, R. J. Knops and L.E. Payne, Decay estimates for the constrained elastic cylinder of variable cross section, Quart. Appl. Math., XLVII, (1989) 325-350.
FR:96 J. N. Flavin and S. Rionero. Qualitative Estimates for Partial Differential Equations: An Introduction, CRC Press, Boca Raton (1996).
HS M. Hayes and G. Saccomandi. Finite amplitude transverse waves in special incompressible viscoelastic solids, J. of Elasticity, 59, (2000) 213-225.
H D. D. Holm, V. Putkaradze. P. D. Weidman and B. A. Wingate. Boundary effects on exact solutions of the Lagrangian-averaged Navier-Stokes-$\alpha$ equations, J. of Statistical Physics, 113, (2003) 841-854.
HOR:89 C.O. Horgan. Recent developments concerning Saint-Venant’s Principle: An update, Applied Mechanics Reviews, 42, (1989) 295-303.
HOR:96b C.O. Horgan. Recent developments concerning Saint-Venant’s Principle: A second update, Applied Mechanics Reviews, 49, (1996) 101-111.
HOR-KNOW:83 C.O. Horgan and J.K. Knowles. Recent developments concerning Saint-Venant’s Principle, Advances in Applied Mechanics (Ed. J. W. Hutchinson and T.Y. Wu), 23, Academic Press, New York (1983) 179-269.
Jordan P. M. Jordan and C. Feuillade. On the propagation of transient acoustic waves in isothermal bubbly liquids, Physics Letters A, 350 (2006) 56-62.
LMA M. Lazar, G. A. Maugin and E.C. Aifantis. On a theory of nonlocal elasticity of bi-Helmholtz type and some applications, International Journal of Solids and Structures, 43 (2006) 1404-1421.
LQ: M.C. Leseduarte and R. Quintanilla. Some qualitative properties of solutions of the system governing acoustic waves in bubbly liquids. International Journal of Engineering Science, to appear (2006).
LOV A. E. H. Love. A Treatise on the Mathematical Theory of Elasticity. Dover Publications. New York, fourth edition (1944).
Mau G. A. Maugin. Nonlinear Waves in Elastic Crystals. Oxford University Press. Oxford, (1999).
MQ:05 J. Muñoz-Rivera and R. Quintanilla. On the time polynomial decay in elastic solids with voids. Manuscript (2005).
PVM A. F. Pazoto, J. C. Vila Bravo and J. E. Muñoz-Rivera. Asymptotic stability of semigroups associated to linear weak dissipative systems, Math. Computer Modelling, 40, (2004), 387-392.
RRG M. B. Rubin, P. Rosenau and O. Gottlieb. Continuum model of dispersion caused by an inherent material characteristic length, J. Appl. Phys. 77, (1995) 4054-4063.
Sax R. Saxton. The Cauchy and backward-Cauchy problem for a nonlinear hyperelastic infinite rod, Lectures Notes in Mathematics, 1032, Springer-Verlag. Berlin, (1982) 427-448.
villa P. Villaggio. Mathematical Models for Elastic Structures. Cambridge University Press. Cambridge, (1997).
W L. van Wijngaarden. One-dimensional flow of liquids containing small gas bubbles, Ann. Rev. Fluid Mech., 4, (1972) 369-396.
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Additional Information
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
Keywords:
Phragmén-Lindelöf 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.
