This thesis is concerned with the initial-boundary value problem for a viscoelastic wave equation with acoustic boundary conditions, in which spatially varying viscoelastic coefficients, nonlinear damping, a logarithmic source term, and velocity-delay feedback are simultaneously incorporated. A finite-time blow-up criterion for solutions is established. Most existing studies are restricted to constant-coefficient media, models without time delay, or systems involving a single nonlinear effect, which are inadequate for describing the realistic dynamical behavior of gradient composite materials and acoustically coupled structures. To overcome these limitations, the present model integrates four complex mechanisms, namely variable coefficients, time delay, logarithmic nonlinearity, and acoustic boundary conditions, into a unified framework. By introducing a suitable delay transformation variable, the original integro-differential equation is reformulated into an equivalent system. Under appropriate assumptions on the memory kernel and spatially varying coefficients, the classical energy method is refined, and two auxiliary functionals adapted to the multiple coupled effects are constructed to derive delicate estimates. It is proved that every nontrivial solution blows up in finite time whenever the initial energy is negative. Moreover, explicit upper and lower bounds for the blow-up time are rigorously derived.