Well posedness and energy estimate of a truncated porous thermoelastic system with a memory thermal law

This work investigates the well-posedness and long-time behavior of solutions to a truncated porous elastic system coupled with the heat equation, which is modeled by the memory-type Gurtin-Pipkin thermal law. The system is free from the adverse effects associated with the second spectrum of frequencies. We employ a non-classical approach based on operator theory to demonstrate the existence of a unique solution. Additionally, we utilize the multiplier method to establish the exponential decay of the solution, independent of any relationships between the coefficients of the system. Our findings significantly enhance the results presented in [Fareh A. Exponential stability of a porous thermoelastic system with Gurtin-Pipkin thermal law. RACSAM. 2022;116(1):6., Hao J, Yang J. Exponential stability for porous thermoelastic system with Gurtin-Pipkin flux. Electron J Differ Equations. 2023;2023(44):1–17.] for classical porous thermoelastic problems.