On the time decay for the MGT-type porosity problems

Jacobo Baldonedo; José R. Fernández; Ramón Quintanilla

doi:10.3934/dcdss.2022009

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2022, Volume 15, Issue 8: 1941-1955. Doi: 10.3934/dcdss.2022009

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On the time decay for the MGT-type porosity problems

1.
CINTECX, Departamento de Ingeniería Mecánica, Universidade de Vigo, Campus As Lagoas Marcosende s/n, 36310 Vigo, Spain
2.
Departamento de Matemática Aplicada I, Universidade de Vigo, Escola de Enxeñería de Telecomunicación, Campus As Lagoas Marcosende s/n, 36310 Vigo, Spain
3.
Departamento de Matemáticas, E.S.E.I.A.A.T.-U.P.C., Colom 11, 08222 Terrassa, Barcelona, Spain

^*Corresponding author: Ramón Quintanilla
^*Corresponding author: Ramón Quintanilla

Received: September 30, 2021

Revised: November 30, 2021

Published: August 2022

This paper is part of the projects PGC2018-096696-B-I00 and PID2019-105118GB-I00, funded by the Spanish Ministry of Science, Innovation and Universities and FEDER "A way to make Europe". The authors would also like to acknowledge the comments provided by the reviewer, which helped to improve the final quality of the paper

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Abstract

In this work we study three different dissipation mechanisms arising in the so-called Moore-Gibson-Thompson porosity. The three cases correspond to the MGT-porous hyperviscosity (fourth-order term), the MGT-porous viscosity (second-order term) and the MGT-porous weak viscosity (zeroth-order term). For all the cases, we prove that there exists a unique solution to the problem and we analyze the resulting point spectrum. We also show that there is an exponential energy decay for the first case, meanwhile for the second and third case only a polynomial decay is found. Finally, we present some one-dimensional numerical simulations to illustrate the behaviour of the discrete energy for each case.

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Mathematics Subject Classification: Primary: 74F10, 35B40; Secondary: 65M60.

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Figure 1. Roots behaviour for the fourth-order dissipation mechanism

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Figure 2. Roots behaviour for the second-order dissipation mechanism

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Figure 3. Roots behaviour for the zero-order dissipation mechanism

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Figure 4. Example 1: Dependence of the solution with respect to parameter $ \kappa $ (fourth-order dissipation mechanism- Case (i))

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Figure 5. Example 2: Dependence of the solution with respect to parameter $ a $ (second-order dissipation mechanism- Case (ii))

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Figure 6. Example 3: Dependence of the solution with respect to parameter $ \xi $ (zeroth-order dissipation mechanism- Case (iii))

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