Decay estimates for the constrained elastic cylinder of variable cross section

Flavin, J. N.; Knops, R. J.; Payne, L. E.

doi:10.1090/qam/998106

Cited by 113 publications

(82 citation statements)

References 16 publications

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“…Such studies, clearly related to Saint-Venant's principle, are significant for investigations into boundary effects and similar phenomena. In elasticity, both growth and decay estimates have been derived for nonprismatic and prismatic finite and semi-infinite cylinders (e.g., [1,2]), and for cone-like regions (e.g., [5]) unbounded in a given direction. The method of derivation, however, does not readily extend to thermoelasticity, although the cylinder has been successfully treated by separate weighting of the mechanical and thermal energies [6].…”

Section: Introductionmentioning

confidence: 99%

Spatial Behaviour in Thermoelastostatic Cylinders of Indefinitely Increasing Cross-Section

Knops

Quintanilla²

2015

J Elast

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Alternative growth and decay estimates, reminiscent of the classical Phragmén-Lindelöf principle, are derived for a linearised thermoelastic body whose plane cross-sections increase unboundedly with respect to a given direction. The proof uses a modified Poincaré inequality to construct a differential inequality for a weighted linear combination of the cross-sectional mechanical and thermal energy fluxes. Decay estimates are deduced also for the cross-sectional mean square measures of the displacement and temperature. An explicit upper bound in terms of base data is established for the amplitude occurring in the decay estimates.

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Section: Introductionmentioning

confidence: 99%

Spatial Behaviour in Thermoelastostatic Cylinders of Indefinitely Increasing Cross-Section

Knops

Quintanilla²

2015

J Elast

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“…Similar theorems may be obtained for a larger class of domains by means of the techniques used in [11], provided that the functions pi satisfy the conditions (1 .6) . A slight change of hypotheses permits the previous method still to be applicable .…”

Section: Decay Of the Energymentioning

confidence: 59%

Some theorems of Phragmen-Lindelof type for nonlinear partial differential equations

Quintanilla¹

1993

Publicacions Matemàtiques

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SOME THEOREMS OF PHRAGMEN-LINDELOF TYPE FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS RAMÓN QUINTANILLAThe present paper studies second order partial differential equations in two independent variables of the form Div(pl ¡u,, J' -lu,l , P21U,2 I n-1 u,2) = 0. We obtain decay estimates for the solutions in a semi-infinite strip. The results may be seen as theorems of Phragmen-Lindelof type. The method is strongly based on the ideas of Horgan and Payne [5], [6], [8] .

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“…This is because (3.6) implies that 10) and from these statements we obtain an alternative of Phragmén-Lindelöf type which states (see [5]) that the solutions either grow exponentially for z sufficiently large with the measure…”

Section: Phragmén-lindelöf Alternativementioning

confidence: 78%

Spatial behavior for a fourth-order dispersive equation

Quintanilla

Saccomandi

2006

Quart. Appl. Math.

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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. Introduction.In recent years much attention has been directed to the study of the damping of end effects in several thermomechanical situations and to the general study of the spatial behavior of solutions of partial differential equations and systems. The history and development of these questions was extensively surveyed by Horgan and Knowles [11] in a work periodically updated by Horgan [9,10]. Moreover, the books by Ames and Straughan [1], Flavin and Rionero [6] and Antontsev et al. [2] are devoted to such matters. The energy method is widely used to study the spatial behavior of solutions of partial differential systems.In this note we investigate the spatial behavior of a linear equation of fourth order which is encountered in several frameworks when we take into account dispersive effects.

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Decay estimates for the constrained elastic cylinder of variable cross section

Cited by 113 publications

References 16 publications

Spatial Behaviour in Thermoelastostatic Cylinders of Indefinitely Increasing Cross-Section

Spatial Behaviour in Thermoelastostatic Cylinders of Indefinitely Increasing Cross-Section

Some theorems of Phragmen-Lindelof type for nonlinear partial differential equations

Spatial behavior for a fourth-order dispersive equation

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