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)‖)4u+ ∫ t 0 h(t− τ)div[a(x)∇u(τ)]dτ +|u|u+ 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.

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