Asymptotic behavior of the porous elastic system with dual phase lag model: Classical versus second spectrum perspectives

This paper aims to analyze the energy decay of the thermoelastic porous system. The dual-phase lag theory is used to model heat transfer. We consider two perspectives: the classical approach and the second spectrum approach. For the classical approach, the well-posedness is obtained via the semigroup theory and the system is exponentially stable under equal wave speed conditions. On the opposite, we show a polynomial decay. On the other hand, the well-posedness of the truncated system is obtained via the Faedo Galerkin method, and the system is exponentially stable without any assumptions on the physical parameters.