Stability of a Transmission Problem for a Shear Beam Model

Abstract

ABSTRACT In this paper, we consider a transmission problem for a shear beam model consisting of two different materials joined at an interface. The system is modeled by coupled hyperbolic‐elliptic equations where linear frictional damping is applied to the transverse displacement. Using semigroup theory and nonclassical differential operators, we prove the existence and uniqueness of solutions. We show that the system is exponentially stable when damping is distributed globally across both segments. However, in the case of localized damping on only one segment, we prove that the system is strongly stable but lacks semi‐uniform stability, failing to achieve either exponential or polynomial decay.