Energy decay analysis for Porous elastic system with thermoelasticity of type III: A second spectrum approach

Numerous studies have been conducted to investigate porous systems under different damping effects. Recent research has consistently achieved the expected exponential decay of energy solutions when employing stabilization techniques that involve non-physical assumptions of equal wave velocities. In this study, we examine a one-dimensional thermoelastic porous system within the framework of the second frequency spectrum. Remarkably, we demonstrate that exponential decay can be achieved without relying on the assumption of equal wave speeds. We consider the porous system, and we incorporated thermoelastic damping based on the Green–Naghdi law of heat conduction into our study. To begin with, we use the Faedo–Galerkin approximation method to validate the global well-posedness of the system. By utilizing a Lyapunov functional, we establish exponential stability without relying on the assumption of equal wave speed. We then introduce and analyze a numerical scheme. Finally, by assuming additional regularity of the solution, we derive a priori error estimates.