Decay and growth estimates for solutions of second-order and third-order differential-operator equations

Kalantarov, Varga K.; Yilmaz, Yalcin

doi:10.1016/j.na.2013.04.016

Cited by 5 publications

(10 citation statements)

References 7 publications

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“…See [8] and [13] and their references. Other recent studies on this equation can be seen in [9] and [11], for example. The same equation appears in viscoelasticity theory, then under the name of Standard Linear Viscoelastic model.…”

Section: Marta Pellicer and Joan Solà-moralesmentioning

confidence: 99%

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Optimal scalar products in the Moore-Gibson-Thompson equation

Pellicer¹,

Solà-Morales²

2019

Evolution Equations &Amp; Control Theory

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We study the third order in time linear dissipative wave equation known as the Moore-Gibson-Thompson equation, that appears as the linearization of a the Jordan-Moore-Gibson-Thompson equation, an important model in nonlinear acoustics. The same equation also arises in viscoelasticity theory, as a model which is considered more realistic than the usual Kelvin-Voigt one for the linear deformations of a viscoelastic solid. In this context, it is known as the Standard Linear Viscoelastic model. We complete the description in [13] of the spectrum of the generator of the corresponding group of operators and show that, apart from some exceptional values of the parameters, this generator can be made to be a normal operator with a new scalar product, with a complete set of orthogonal eigenfunctions. Using this property we also obtain optimal exponential decay estimates for the solutions as t → ∞, whether the operator is normal or not.

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Section: Marta Pellicer and Joan Solà-moralesmentioning

confidence: 99%

“…Observe that, as µ n > 0, the sign of (19) is determined by the sign of the second degree polynomial d(µ n ). The roots of this quadratic equation are the constants m 1 , m 2 defined in (8) with the constants C 1 , C 2 defined in (9). It can be seen that C 2 = 0 if and only of β α = 1 or β α = 9.…”

mentioning

confidence: 99%

Optimal scalar products in the Moore-Gibson-Thompson equation

Pellicer¹,

Solà-Morales²

2019

Evolution Equations &Amp; Control Theory

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“…See [7] and [9] and their references. Other recent studies on this equation can be seen in [2] and [8], for example. The paper [3] deals with the case α/β > 1.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…where σ max = σ max (A) < 0 is the supremum respect to n (which sometimes is a maximum) of the real parts of the solutions of the characteristic equation (8), that is, the real part of the sometimes called dominant spectrum of A (see Proposition 5 below for a description of σ max ). ii) This decay is optimal in the sense that for each ω < σ max there exist solutions U(t) such that U(t) G e −ωt → ∞ as t → ∞.…”

Section: Theoremmentioning

confidence: 99%

Optimal scalar products in the Standard Linear Viscoelastic Model

Pellicer,

Solà-Morales

2015

Preprint

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We study the third order in time linear dissipative wave equation known as the Standard Linear Viscoelastic Model, that appears also as the linearization of the so-called Moore-Gibson-Thompson equation in Nonlinear Acoustics. We complete the description in [9] of the spectrum of the generator of the corresponding group of operators and show that, apart from some exceptional values of the parameters, this generator can be made to be a normal operator with a new scalar product, with a complete set of orthogonal eigenfunctions. Using this property we also obtain sharper decay estimates for the solutions as t → ∞, both when the operator is normal or not.

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“…In this article, in contrast to the works on homogenous equations mentioned in [2][3][4], a non-homogeneous differential-operator equation of the third order has been investigated. It has been shown that every solution to the equation is asymptotically stable, provided that a condition on the function on the right-hand side is established.…”

mentioning

confidence: 99%