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

TL;DR

This paper investigates the slow energy decay rates of solutions to the thermoelastic Bresse system with heat conduction, revealing the influence of wave speeds and initial data regularity on decay behavior.

Contribution

It establishes the decay rate of solutions as $(1+t)^{-1/8}$ for the thermoelastic Bresse system with heat conduction, highlighting the regularity-loss phenomenon and using Fourier-based energy methods.

Findings

Solutions decay at rate $(1+t)^{-1/8}$ in $L^2$-norm.

Decay rate depends on initial data regularity and wave speeds.

Decay is very slow due to regularity-loss phenomenon.

Abstract

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.