Decay Rates of the Solutions to the Thermoelastic Bresse System of Types I and III

In this paper, we study the energy decay for the thermoelastic Bresse system in the whole line with two different dissipative mechanism, 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 $(1+t)^{-\frac{1}{8}}$ in the $L^2$-norm, whenever the initial data belongs to $L^1(R) \cap H^{s}(R)$ 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 \textit{regularity-loss}. The main tool used to prove our results is the energy method in the Fourier space.