Decay estimates for the Cauchy problem for the damped extensible beam equation

Abstract

The extensible beam equation proposed by Woinowsky-Krieger is a fourth order dispersive equation with nonlocal nonlinear terms. In this paper we study the Cauchy problem of the extended model by Ball who proposed the following model with external and structural damping terms:

For this represents a Kelvin–Voigt damping. We show the unique global existence of solutions for this problem and give a precise description of the decay of solutions in time.

Keywords:

• decay estimate
• extensible beam
• Cauchy problem
• Ball’s model
• Kelvin–Voigt damping

AMS Subject Classifications:

• 35G10
• 35B65
• 35G25
• 35E05

Acknowledgements

The article was written when the second author visited the University of Konstanz between April 2014 and February 2015. He expresses his sincere gratitude to the kind hospitality of the people at the University of Konstanz.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The second author is partially supported by the Japan Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (C) [No. 25400172].