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

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.

Keywords

• Decay rate
• heat conduction
• Bresse system
• thermoelasticity