# ASYMPTIOTIC BEHAVIOR FOR THE VISCOELASTIC KIRCHHOFF TYPE EQUATION WITH AN INTERNAL TIME-VARYING DELAY TERM

Kim, Daewook

• Accepted : 2016.05.13
• Published : 2016.05.31
• 30 12

#### Abstract

In this paper, we study the viscoelastic Kirchhoff type equation with the following nonlinear source and time-varying delay $$u_{tt}-M(x,t,{\parallel}{\nabla}u(t){\parallel}^2){\Delta}u+{\int_{0}^{t}}h(t-{\tau})div[a(x){\nabla}u({\tau})]d{\tau}\\+{\parallel}u{\parallel}^{\gamma}u+{\mu}_1u_t(x,t)+{\mu}_2u_t(x,t-s(t))=0.$$ 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.

#### Keywords

viscoelastic Kirchhoff type equation;internal time-varying delay;energy decay rate;energy functional;smallness condition

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#### Cited by

1. Global existence of solutions to a viscoelastic non-degenerate Kirchhoff equation pp.1563-504X, 2018, https://doi.org/10.1080/00036811.2018.1544621