Spatial estimates for an equation with a delay term

Quintanilla, R.

doi:10.1007/s00033-009-0049-4

Cited by 7 publications

(5 citation statements)

References 27 publications

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“…Theorem Assume that conditions (1), (11) and (12) on the elasticity tensor and (2) on the mass density are satisfied. Let {u i } be a solution of the problem determined by system (7), boundary conditions (8), homogeneous initial conditions (10) and asymptotic assumption (13).…”

Section: T)mentioning

confidence: 99%

“…where C 1 is given at (11) and C(D) is the Poincaré constant for the domain D. It is worth noting that…”

Section: Spatial Estimatesmentioning

confidence: 99%

“…Several decay estimates for elliptic [4], parabolic [6,7] and hyperbolic equations [2,5] have been obtained. A spatial decay estimate for an equation with delay has recently been obtained [11]. In these contributions the authors obtain growth/decay estimates for the solutions.…”

Section: Introductionmentioning

confidence: 99%

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Polynomial and Exponential Spatial Decay Estimates in Elasticity

Quintanilla¹

2008

J Elasticity

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In this note we investigate the spatial behavior of the solutions of a combination of a hyperbolic system with an elliptic system. We consider a semi-infinite cylinder which is the union of two sub-cylinders. In one of them, we assume an elastodynamical problem and in the other an elastostatic problem. Both are coupled through an interface. It is known that the elastostatic problem and the elastodynamic problem have a fast decay (at least exponential). However, as their spatial behaviors are of different kind, it is not clear how this combination could be controlled in a similar way. We prove that the decay of solutions can be controlled in a polynomial way. We also describe how to obtain an upper bound for the amplitude term. We conclude the paper sketching the exponential decay behavior for the harmonic vibrations.

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Section: T)mentioning

confidence: 99%

“…where C 1 is given at (11) and C(D) is the Poincaré constant for the domain D. It is worth noting that…”

Section: Spatial Estimatesmentioning

confidence: 99%

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Polynomial and Exponential Spatial Decay Estimates in Elasticity

Quintanilla¹

2008

J Elasticity

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“…A complete explanation about the uniqueness and the stability analysis of the solutions of the phase-lagging equations can be found in [116][117][118][119][120][121]. Also, the stability analysis is performed for the TPL model by [122,123] and for the three-dual-phase-lag model (TDPL) by [124,125].…”

Section: Stabilitymentioning

confidence: 99%

Macro- to Nanoscale Heat and Mass Transfer: The Lagging Behavior

2015

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The classical model of the Fourier's law is known as the most common constitutive relation for thermal transport in various engineering materials. Although the Fourier's law has been widely used in a variety of engineering application areas, there are many exceptional applications in which the Fourier's law is questionable. This paper gathers together such applications. Accordingly, the paper is divided into two parts. The first part reviews the papers pertaining to the fundamental theory of the phase-lagging models and the analytical and numerical solution approaches. The second part wrap ups the various applications of the phase-lagging models including the biological materials, ultra-high-speed laser heating, the problems involving moving media, micro/nanoscale heat transfer, multi-layered materials, the theory of thermoelasticity, heat transfer in the material defects, the diffusion problems we call as the non-Fick models, and some other applications. It is predicted that the interest in the field of phase-lagging heat transport has grown incredibly in recent years because they show good agreement with the experiments across a wide range of length and time scales.

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“…Decay estimates for elliptic [1], parabolic [2,3], hyperbolic [4][5][6] equations and/or combination of them [7,8] have been obtained in this period. However, as far as the authors know, there are only two contributions [9,10] to the case of partial differential equations with a delay term. In this paper we provide a new contribution to study the case of a partial differential equation with two delay terms.…”

Section: Introductionmentioning

confidence: 99%

Spatial behavior for solutions in heat conduction with two delays

Leseduarte

Quintanilla

2014

Applicable Analysis

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In this note we investigate the spatial behavior of the solutions of the equation proposed to describe a theory for the heat conduction with two delay terms. We obtain an alternative of the Phragmén-Lindelöf type, which means that the solutions either decay or blow-up at infinity, both options in an exponential way. We also describe how to obtain an upper bound for the amplitude term. This is the first contribution on spatial behavior for partial differential equations involving two delay terms. We use energy arguments. The main point of the contribution is the use of an exponentially weighted energy function.

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Spatial estimates for an equation with a delay term

Cited by 7 publications

References 27 publications

Polynomial and Exponential Spatial Decay Estimates in Elasticity

Polynomial and Exponential Spatial Decay Estimates in Elasticity

Macro- to Nanoscale Heat and Mass Transfer: The Lagging Behavior

Spatial behavior for solutions in heat conduction with two delays

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