Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 30, pp. 1-19.

Title: Exponential stability of a damped beam-string-beam transmission problem
       
Authors: Bienvenido Barraza Martinez (Univ. del Norte, Barranquilla, Colombia)
         Jairo Hernandez Monzon (Univ. del Norte, Barranquilla, Colombia)
         Gustavo Vergara Rolong (Univ. del Norte, Barranquilla, Colombia)

Abstract: 
We consider a beam-string-beam transmission problem, where two structurally
damped or undamped beams are coupled with a frictionally damped string by
transmission conditions. We show that for this structure, the
dissipation produced by the frictional part is strong enough to produce
exponential decay of the solution no matter how small is its size.
For the exponential stability in the damped-damped-damped situation
we use energy method. For the undamped-damped-undamped situation
we use a frequency domain method from semigroups theory,
which combines a contradiction argument with the multiplier technique
to carry out a special analysis for the resolvent.
Additionally, we show that the solution first defined by the weak
formulation has higher Sobolev space regularity.

Submitted September 30, 2021. Published April 13, 2022.
Math Subject Classifications: 35M33, 35B35, 35B40, 93D23.
Key Words: Exponential stability; transmission problems; frictional damping; 
beam-string equations.