Dynamic behaviors for the acoustic model with variable coefficients and nonautonomous damping

We study the dynamic behaviors of solutions to wave equations with variable coefficients, localized nonautonomous damping, and nonlinear force terms, subject to acoustic boundary conditions. The localized damping can gradually vanish over time. The nonautonomous localized damping and nonlinear force terms cause challenges, resulting in classical approaches handling the nonautonomous low-order term do not work here directly. To address this, we introduce novel techniques. First, assuming a new suitable condition on the nonautonomous term and using Riemannian geometry theory, we establish refined estimates for the lower-order nonautonomous term and a weighted energy sequence relation. Secondly, we introduce a general theory by which one can eventually establish the continuous energy decay rate from the weighted discrete energy sequence relation. This theory can be applied to study the stability of the general system with nonautonomous damping.