Electron. J. Differential Equations, Vol. 2022 (2022), No. 30, pp. 1-19.

Exponential stability of a damped beam-string-beam transmission problem

Bienvenido Barraza Martinez, Jairo Hernandez Monzon, Gustavo Vergara Rolong

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. DOI: https://doi.org/10.58997/ejde.2022.30

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Bienvenido Barraza Martínez
Universidad del Norte
Departamento de Matemáticas y Estadística
Barranquilla, Colombia
email: bbarraza@uninorte.edu.co
Jairo Hernández Monzón
Universidad del Norte
Departamento de Matemáticas y Estadística
Barranquilla, Colombia
email: jahernan@uninorte.edu.co
Gustavo Vergara Rolong
Universidad del Norte
Departamento de Matemáticas y Estadística
Barranquilla, Colombia
email: gvergaraa@uninorte.edu.co

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