Optimal decay for a wave transmission problem with fading memory

Abstract

The transmission of waves over a body composed of two different materials are considered in this paper. One component is a simple elastic part while the other is a viscoelastic component with time-varying delay and damping effect. Besides, one part of the boundary is controlled by a nonlinear memory source term while the other part of the boundary is clamped. Under the influence of time-varying delay and damping effect, we derive a general decay estimate of the energy when the internal viscoelastic memory and the boundary memory decay simultaneously. Moreover, we present some examples to illustrate that the decay rate of the energy is optimal in some range of decay exponent of the memory.

COMMUNICATED BY:

• S. Hansen

Keywords:

• transmission problem
• time-varying delay
• boundary memory source
• general decay
• optimal decay

2010 MSCs:

• 35B40
• 35L53
• 93D15
• 93D20

Acknowledgments

This work is supported by the National Natural Science Foundation of China (No.11671188) and the Fundamental Research Funds for the Central Universities (No.201861002, No.201964008). The authors would like to deeply thank all the reviewers for their insightful and constructive comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by the National Natural Science Foundation of China [grant number 11671188] and the Fundamental Research Funds for the Central Universities [grant numbers 201861002, 201964008].