Exponential stability for wave equation with delay and dynamic boundary conditions

Abstract

In this paper, we consider a multi-dimensional wave equation with delay and dynamic boundary conditions, related to the Kelvin–Voigt damping. By using the Faedo–Galerkin approximations together with some priori estimates, we prove the local existence of solution. Since the damping may stabilize the system while the delay may destabilize it, we discuss the interaction between the damping and the delay, and obtain that the system is uniformly stable when the effect of damping is greater than that of time delay. Exponential stability result of system is also established by constructing suitable Lyapunov functionals.

Keywords:

• Wave equation
• Kelvin–Voigt damping
• energy decay
• dynamic boundary conditions
• delay

AMS Subject Classifications:

• 35L20
• 35B35
• 93D20

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially supported by NNSF of China [grant number 11871315, 61374089]; Natural Science Foundation of Shanxi Province of China [grant number 201801D121003, 201901D111021].