Long-time dynamics of the Kirchhoff equation with variable coefficient rotational inertia and memory

The Kirchhoff model stems from the vibration problem of stretchable strings. This paper investigates the Kirchhoff equation incorporating variable coefficient rotational inertia and memory. By employing the Faedo–Galerkin method, the existence and uniqueness of the solution are established. Moreover, the existence of a global attractor is demonstrated through the proof of a bounded absorbing set and the asymptotic smoothness of the semigroup. The study innovatively explores the long-time dynamical behavior of the Kirchhoff model under the combined effects of variable coefficient rotational inertia, memory, and thermal interactions, thereby extending the model’s theoretical framework. These results provide a robust theoretical foundation for future applications and research endeavors.