Decay rates for elastic-thermoelastic star-shaped networks

Zhong-Jie Han; Enrique Zuazua

doi:10.3934/nhm.2017020

Article Contents

2017, Volume 12, Issue 3: 461-488. Doi: 10.3934/nhm.2017020

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Decay rates for elastic-thermoelastic star-shaped networks

Zhong-Jie Han^1, and
Enrique Zuazua^2,3,4,

1.
School of Mathematics, Tianjin University, 300354 Tianjin, China
2.
DeustoTech -Fundación Deusto, Avda. Universidades, 24, 48007 Bilbao, Basque Country, Spain
3.
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
4.
Facultad Ingeniería, Universidad de Deusto, Avda. Universidades, 24, 48007 Bilbao, Basque Country, Spain

Received: August 31, 2016

Revised: June 30, 2017

Published: October 2017

The first author was supported by the Natural Science Foundation of China grant NSFC-61573252 and China Scholarship Council. The second author was supported by the Advanced Grant DYCON of the European Research Council Executive Agency, ICON of the ANR-2016-ACHN-0014-01 (France), FA9550-14-1-0214 of the EOARD-AFOSR, FA9550-15-1-0027 of AFOSR, the MTM2014-52347 Grant of the MINECO and a Humboldt Research Award at the University of Erlangen-Nürnberg.

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Abstract

This work discusses the asymptotic behaviour of a transmission problem on star-shaped networks of interconnected elastic and thermoelastic rods. Elastic rods are undamped, of conservative nature, while the thermoelastic ones are damped by thermal effects. We analyse the overall decay rate depending of the number of purely elastic components entering on the system and the irrationality properties of its lengths.

First, a sufficient and necessary condition for the strong stability of the thermoelastic-elastic network is given. Then, the uniform exponential decay rate is proved by frequency domain analysis techniques when only one purely elastic undamped rod is present. When the network involves more than one purely elastic undamped rod the lack of exponential decay is proved and nearly sharp polynomial decay rates are deduced under suitable irrationality conditions on the lengths of the rods, based on Diophantine approximation arguments. More general slow decay rates are also derived. Finally, we present some numerical simulations supporting the analytical results.

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Mathematics Subject Classification: Primary: 35B40, 93D20; Secondary: 35M10, 74F05.

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Figure 1. Transmission problem in 1-d elasticity-thermoelasticity

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Figure 2. Star-shaped thermoelastic-elastic network

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Figure A-1. $u_1(x,t)$

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Figure A-2. $u_2(x,t)$

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Figure A-3. $u_3(x,t)$

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Figure A-4. $\theta_1(x,t)$

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Figure A-5. $\theta_2(x,t)$

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Figure A-6. $\theta_3(x,t)$

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Figure B-1. $u_1(x,t)$

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Figure B-2. $u_2(x,t)$

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Figure B-3. $u_3(x,t)$

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Figure C-1. $u_1(x,t)$

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Figure C-2. $u_2(x,t)$

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Figure C-3. $u_3(x,t)$

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Figure C-4. $\theta_1(x,t)$

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Figure C-5. $\theta_2(x,t)$

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Figure D-1. $u_1(x,t)$

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Figure D-2. $u_2(x,t)$

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Figure D-3. $u_3(x,t)$

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Figure D-4. $\theta_1(x,t)$

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Figure E-1. $u_1(x,t)$

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Figure E-2. $u_2(x,t)$

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Figure E-3. $u_3(x,t)$

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Figure E-4. $\theta_1(x,t)$

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Figure F-1. Logarithmic scale of energy for Case A, B, C

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Figure F-2. Energy for Case D, E

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