Long-time behaviors of wave equations stabilized by boundary memory damping and friction damping

Abstract

In this paper, we study the long-time behaviors of wave equations subject to boundary memory damping and friction damping. Different from assumptions that memory kernel is a nonnegative, monotone function in the previous literatures, we assume that the primitive of the memory kernel is a generalized positive definite kernel (abbreviated to GPDK), which may vary sign or oscillate. The key to the problem lies in establishing the connection between memory damping and energy terms. By combining the properties of the positive definite kernel with classical multiplier methods, and constructing auxiliary systems, we ultimately establish the asymptotic stability, exponential stability and polynomial stability of systems featuring boundary memory damping and friction damping. To illustrate our theoretical results, we provide some numerical simulations.