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STABILIZATION FOR THE VISCOELASTIC KIRCHHOFF TYPE EQUATION WITH A NONLINEAR SOURCE

Kim, Daewook

  • Received : 2016.01.08
  • Accepted : 2016.01.30
  • Published : 2016.01.30

Abstract

In this paper, we study the viscoelastic Kirchhoff type equation with a nonlinear source $$u^{{\prime}{\prime}}-M(x,t,{\parallel}{\bigtriangledown}u(t){\parallel}^2){\bigtriangleup}u+{\int}_0^th(t-{\tau})div[a(x){\bigtriangledown}u({\tau})]d{\tau}+{\mid}u{\mid}^{\gamma}u=0$$. 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;energy decay rate;energy functional;smallness condition

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Acknowledgement

Supported by : National Research Foundation of Korea (NRF)