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

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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