Exponential attractor for Kirchhoff model with time delay and thermal effect

Abstract

The Kirchhoff model is derived from the vibration problem of stretchable strings. In this paper, we focus on the long-time dynamics of the Kirchhoff model with time delay and thermal effect. We first proved the existence and uniqueness of the solution through the Faedo–Galerkin method and prior estimates. After that, it is further proved that the dynamical system has a bounded absorption set, and it is also verified that the system is quasi-stable on the bounded absorbing set. Finally, the existence of exponential attractor in dynamical system is proved. This innovatively considers the long-time dynamical behavior of Kirchhoff model under the simultaneous action of variable coefficient, thermal effect and time delay comprehensively. It also promotes the relevant conclusions of the Kirchhoff model. The findings of this paper provide a theoretical basis for subsequent application and research.