Exponential Stabilization of an Euler-Bernoulli Beam by a Heat Equation Involving Memory Effects

This study delves into the dynamic behavior of a coupled system comprising an Euler-Bernoulli beam equation and a heat equation with memory, governed by different heat conduction laws. These laws, represented by a parameter m include the Coleman-Gurtin and Gurtin-Pipkin laws, which uniquely influence the material's temperature evolution. By formulating a set of partial differential equations along with appropriate boundary and transmission conditions, we lay the groundwork for analyzing the coupled system's behavior. Our investigation primarily centers on understanding the asymptotic properties of the solution semigroup within the framework of Dafermos history space. We establish exponential stability under both the Coleman-Gurtin and Gurtin-Pipkin heat conduction models by imposing distinct conditions on the relaxation function. The article concludes with a comparison of the two heat conduction laws and a discussion on potential directions for extending this research.