Abstract
We examine the Cauchy problem for a model of linear acoustics, called the Moore–Gibson–Thompson equation, describing a sound propagation in thermo-viscous elastic media with two temperatures on cylindrical domains. For an adequate combination of the parameters of the model we prove Lp-Lq-well-posedness, and we provide maximal regularity estimates which are optimal thanks to the theory of operator-valued Fourier multipliers.
Highlights
IntroductionThe usage of operator-valued Fourier multipliers to treat cylindrical domains was first carried out in [21] in a Besov-space setting
We consider the Moore–Gibson–Thompson equation with two temperatures in a cylindrical domain Ω = U × V ⊂ Rn+d endowed with Dirichlet boundary conditions:
The model (1) has been recently proposed by Quintanilla [1]. This equation considers two distinct temperatures acting on the heat conduction by means of the Moore–Gibson–Thompson equation: The conductive temperature and thermodynamic temperature
Summary
The usage of operator-valued Fourier multipliers to treat cylindrical domains was first carried out in [21] in a Besov-space setting. In that paper the author obtains semiclassical fundamental solutions for a wide variety of elliptic operators on infinite cylindrical domains Rn × V As a result, they succeed obtaining the key for solving related elliptic and parabolic, as well as hyperbolic problems. The main difficulty relies in the verification of the so-called R-boundedness property that must be satisfied by certain sets of operators To overcome this difficulty, we will employ a criteria established by Denk, Hieber, and Prüss in the reference [24] that reduces the problem to the localization of the spectrum of the Laplacian. (i.e., the Moore–Gibson–Thompson equation) as a possible model for the vertical displacement in viscoelastic plates [26]
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