General decay for the viscoelastic wave equation for Kirchhoff-type containing Balakrishnan-Taylor damping, nonlinear damping and logarithmic source term under acoustic boundary conditions

In this paper, we study the general decay result of the viscoelastic wave equation of Kirchhoff-type with Balakrishnan-Taylor damping, weakly nonlinear time-dependent damping, and a logarithmic source term under acoustic boundary conditions. Our analysis is conducted under minimal conditions on the relaxation function g in $L^{1} (0, \infty )$ , specifically, $g'(t) \leq -\mu (t) G(g(t))$ , where G is an increasing and convex function near the origin and μ is a nonincreasing function. Moreover, we derive the energy decay rates that depend on the functions μ, ξ, and G, as well as the function K defined by $k_{0}$ , which characterizes the growth of k at the origin. This result is novel and extends earlier results in the literature.