Asymptiotic Behavior for the Viscoelastic Kirchhoff Type Equation With an Internal Time-Varying Delay Term

In this paper, we study the viscoelastic Kirchhoff type equation with the following nonlinear source and time-varying delay uₜₜ-M(x,t,∥∇u(t)∥²)Δu+∫₀ᵗh(t-τ)div[a(x)∇u(τ) ] dτ\\+∥u∥γu+μ₁uₜ(x,t)+μ₂uₜ(x,t-s(t))=0. Under the smallness condition with respect to Kirchhoff coefficient and the relaxation function and other assumptions, we prove the uniform decay rate of the Kirchhoff type energy.