On the decay rate of solutions of the Bresse system with Gurtin-Pipkin thermal law

We consider the Cauchy problem for the one-dimensional Bresse system coupled with the heat conduction, wherein the latter is described by the Gurtin–Pipkin thermal law. We study the decay properties of the solution using the energy method in the Fourier space (to build an appropriate Lyapunov functional) accompanied with some integral estimates. In fact we prove that the dissipation induced by the heat conduction is very weak and produces very slow decay rates. In addition in some cases, those decay rates are of regularity-loss type. Also, we prove that there is a number (depending on the parameters of the system) that controls the decay rate of the solution and the regularity assumptions on the initial data. In addition, we show that in the absence of the frictional damping, the memory damping term is not strong enough to produce a decay rate for the solution. In fact, we show in this case, despite the fact that the energy is still dissipative, the solution does not decay at all. This result improves and extends several results, such as those in Appl. Math. Optim. (2016), to appear, Communications in Contemporary Mathematics 18(4) (2016), 1550045, Math. Methods Appl. Sci. 38(17) (2015), 3642–3652 and others.