DOI QR코드

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

Kim, Daewook

  • Received : 2016.04.19
  • Accepted : 2016.05.13
  • Published : 2016.05.31

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