uniform energy decay rates for nonlinear viscoelastic wave equation with nonlocal boundary damping

in this paper,weconsider the uniform decay estimates of solutions for the viscoelastic wave equation

u_tt − κ₀δu +∫ ^t₀g(t − s)div [a(x)∇u(s)] ds + b(x)h(u_t ) = 0 in ω × (0,∞).

under weak assumptions on the functions g, h and f ,weprove the energy functional decays exponentially or polynomially to zero as the time goes to infinity by introducing brief lyapunov functions and precise priori estimates.