Electron. J. Differential Equations, Vol. 2017 (2017), No. 73, pp. 1-26.

Decay rates for solutions to thermoelastic Bresse systems of types I and III

Fernando A. Gallego, Jaime E. Munoz Rivera

Abstract:
In this article, we study the energy decay for the thermoelastic Bresse system in the whole line with two dissipative mechanisms, given by heat conduction (Types I and III). We prove that the decay rate of the solutions are very slow. More precisely, we show that the solutions decay with the rate of $(1+t)^{-1/8}$ in the $L^2$-norm, whenever the initial data belongs to $L^1(\mathbb{R}) \cap H^{s}(\mathbb{R})$ for a suitable s. The wave speeds of propagation have influence on the decay rate with respect to the regularity of the initial data. This phenomenon is known as regularity-loss. The main tool used to prove our results is the energy method in the Fourier space.

Submitted February 17, 2016. Published March 15, 2017.
Math Subject Classifications: 35B35, 35L55, 93D20.
Key Words: Decay rate; heat conduction; Bresse system; thermoelasticity.

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Fernando A. Gallego
Centre de Robotique (CAOR), MINES Paristech
PSL Research University, 60 boulevard Saint-Michel
75272 Paris Cedex 06, France
email: ferangares@gmail.com
Jaime E. Muñoz Rivera
Laboratório de Computução Científica, LNCC
Petrópolis, 25651-070, RJ, Brazil
email: rivera@lncc.br, rivera@im.ufrj.br

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