Stability of variable-coefficient wave equations with dynamic boundary condition and boundary memory damping

In this paper, we study the stability of variable-coefficient wave equations with dynamic boundary condition and boundary memory damping. Different with the previous assumptions that the kernel function is decreasing and its derivative can be controlled by itself, we assume the kernel function possesses strongly positive definite primitive, which may oscillate or vary sign. Due to the absence of internal memory damping, it is not possible to directly utilize the properties of the kernel function to estimate the system's potential/kinetic energy terms. Here, the multiplier method is employed to convert the energy estimation into boundary term estimates. By combining Riemannian geometry theory and characteristics of positive definite kernels along with higher-order energy estimate and the construction of auxiliary systems, we eventually obtain the stability of the system.