Dynamic Behavior and Exponential Stability of the Modified Moore–Gibson–Thompson Thermoelastic Model with Frictional Damping

This paper investigates a modified one-dimensional Moore–Gibson–Thompson (MGT) thermoelasticity model that significantly extends the classical formulation by incorporating two key structural modifications: frictional damping and a novel cross-coupling structure. The system introduces a viscous frictional damping mechanism proportional to the velocity acting on the mechanical (elastic) field, enhancing dissipation, which is a common feature in models extending Green–Naghdi Type III thermoelasticity. The core novelty, however, lies in introducing an additional coupling structure that explicitly links the thermal relaxation effects with the mechanical dissipation effects. This modification moves beyond the standard MGT coupling and is rooted in an effort to model complex visco-thermal interactions, representing the primary contribution to the literature. The well posedness of this modified system is first established using semigroup theory. Through the construction of a new Lyapunov functional, sufficient conditions are then rigorously derived, ensuring the exponential stability of solutions under specific parameter regimes. Furthermore, a critical balance condition is identified between the thermal conductivity and the thermal relaxation time, beyond which the system’s energy decay ceases to be exponential. Finally, numerical experiments employing an explicit–implicit finite difference scheme validate the theoretical findings and illustrate the substantial influence of both the modified coupling and the frictional damping on the system’s long-term energy behavior.