Gevrey regularity for the Euler–Bernoulli beam equation with localized structural damping

Abstract

We study a Euler–Bernoulli beam equation with localized discontinuous structural damping. As our main result, we prove that the associated $C_0$-semigroup $(S(t){)}_{t\ge 0}$ is of Gevrey class $\delta >24$ for t>0, hence immediately differentiable. Moreover, we show that $(S(t){)}_{t\ge 0}$ is exponentially stable.

Keywords:

• Euler–Bernoulli beam
• localized structural damping
• Gevrey class
• differentiability
• exponential stability

2010 Mathematics Subject Classifications:

• 35B65
• 35B35
• 47D06
• 74K10

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research of M. Caggio leading to these results has received funding from the Czech Sciences Foundation (GAČR) [grant number 22-01591S]. Moreover, M. Caggio has been supported by the Praemium Academiae of Š. Nečasová. Czech Academy of Sciences (CAS) is supported by the RVO:67985840. F. Dell'Oro is member of Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA), Istituto Nazionale di Alta Matematica.