On the Moore–Gibson–Thompson Equation and Its Relation to Linear Viscoelasticity

Dell’Oro, Filippo; Pata, Vittorino

doi:10.1007/s00245-016-9365-1

Cited by 104 publications

(82 citation statements)

References 11 publications

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“…The critical role of γ for a dissipative behaviour was recently pointed out also in Dell'Oro and Pata [3], within the framework of viscoleasticity. (We add that linear and true nonlinear variants of the MGT equation including an additional memory term have been the object of recent investigation; see [10] and references therein.)…”

Section: Introductionmentioning

confidence: 72%

On the regularity of solutions to the Moore–Gibson–Thompson equation: a perspective via wave equations with memory

Bucci

Pandolfi

2019

J. Evol. Equ.

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We undertake a regularity analysis of the solutions to initial/boundary value problems for the (third-order in time) Moore-Gibson-Thompson (MGT) equation. The key to the present investigation is that the MGT equation falls within a large class of systems with memory, with affine term depending on a parameter. For this model equation a regularity theory is provided, which is of also independent interest; it is shown in particular that the effect of boundary data that are square integrable (in time and space) is the same displayed by wave equations. Then, a general picture of the (interior) regularity of solutions corresponding to homogeneous boundary conditions is specifically derived for the MGT equation in various functional settings. This confirms the gain of one unity in space regularity for the time derivative of the unknown, a feature that sets the MGT equation apart from other PDE models for wave propagation. The adopted perspective and method of proof enables us to attain as well the (sharp) regularity of boundary traces.

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Section: Introductionmentioning

confidence: 72%

On the regularity of solutions to the Moore–Gibson–Thompson equation: a perspective via wave equations with memory

Bucci

Pandolfi

2019

J. Evol. Equ.

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“…In particular, the associated semigroup on the natural weak energy space is exponentially stable if and only if we are in the subcritical regime, corresponding to χ > 0 (see e.g. [12,23,31,39,40]…”

Section: Mgt Heat Conductionmentioning

confidence: 99%

“…where A is a strictly positive operator on some Hilbert space H and α, β, γ > 0 are given parameters, has deserved much attention in recent years, with several papers appeared in the literature on the argument (see [3,11,12,13,23,25,31,39,40], among others). The model has been originally introduced in connection with fluids mechanics [50].…”

Section: Introductionmentioning

confidence: 99%

Thermoelasticity of Moore–Gibson–Thompson type with history dependence in the temperature

Conti

Pata

Quintanilla

2020

ASY

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In this paper, we consider a thermoelastic model where heat conduction is described by the history dependent version of the Moore-Gibson-Thompson equation, arising via the introduction of a relaxation parameter in the Green-Naghdi type III theory. The well-posedness of the resulting integro-differential system is discussed. In the one-dimensional case, the exponential decay of the energy is proved.

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“…Remark When

μ false(s false) = ϰ δ e^{- δ s}

, the set Z turns out to be nonempty if and only if

α < ϰ

. In such a situation, it reduces to the single (real) point

Z = \{- \frac{α δ}{α + ϰ}\} .

Moreover, for every

ζ \in {normalΠ}_{δ}

and

λ \in σ (A)

, it is immediate to check that

λ f false(ζ false) + ζ^{2} = 0 \Leftrightarrow P_{λ} false(ζ false) = 0,

where

P_{λ} false(ζ false)

is the third‐order complex polynomial

ζ^{3} + δ ζ^{2} + λ false(α + ϰ false) ζ + λ α δ = 0 .

Hence, if

α < ϰ

, in the vertical strip

Π_{δ}

the spectrum of

double-struckA

coincides with the one of the infinitesimal generator of the semigroup generated by the Moore–Gibson–Thompson equation

\overset{⃛}{u} false(t false) + δ \ddot{u} false(t false) + false(α + ϰ false) A \ddot{u} false(t false) + α δ A u false(t false) = 0

(see for details).…”

Section: The Main Resultsmentioning

confidence: 99%

On the spectrum of the equation of linear viscoelasticity

Dell’Oro

2018

Mathematische Nachrichten

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On the Moore–Gibson–Thompson Equation and Its Relation to Linear Viscoelasticity

Cited by 104 publications

References 11 publications

On the regularity of solutions to the Moore–Gibson–Thompson equation: a perspective via wave equations with memory

On the regularity of solutions to the Moore–Gibson–Thompson equation: a perspective via wave equations with memory

Thermoelasticity of Moore–Gibson–Thompson type with history dependence in the temperature

On the spectrum of the equation of linear viscoelasticity

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