Abstract: We considered the initial-boundary value problem for a class of variable-coefficient wave equations with logarithmic source terms and viscoelastic terms. Firstly, we discussed the influence of the logarithmic source terms and viscoelastic terms on the long-time dynamic behavior of solutions in inhomogeneous media. Secondly, by constructing a special type of multiplier and combining it with logarithmic Sobolev inequalities, we prove that the energy functional of the system decays exponentially when the initial energy satisfies 0<E(0)<d(d is the well depth).

Key words: wave equation, logarithmic source term, viscoelastic term, variable coefficient