• Title/Summary/Keyword: viscoelastic Kirchhoff type equation

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EXPONENTIAL DECAY FOR THE SOLUTION OF THE VISCOELASTIC KIRCHHOFF TYPE EQUATION WITH MEMORY CONDITION AT THE BOUNDARY

  • Kim, Daewook
    • East Asian mathematical journal
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    • v.34 no.1
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    • pp.69-84
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    • 2018
  • In this paper, we study the viscoelastic Kirchhoff type equation with a nonlinear source for each independent kernels h and g with respect to Volterra terms. 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.

STABILIZATION FOR THE VISCOELASTIC KIRCHHOFF TYPE EQUATION WITH A NONLINEAR SOURCE

  • Kim, Daewook
    • East Asian mathematical journal
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    • v.32 no.1
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    • pp.117-128
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    • 2016
  • 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.

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

  • Kim, Daewook
    • East Asian mathematical journal
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    • v.32 no.3
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    • pp.399-412
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    • 2016
  • 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.

Linear Approximation and Asymptotic Expansion associated to the Robin-Dirichlet Problem for a Kirchhoff-Carrier Equation with a Viscoelastic Term

  • Ngoc, Le Thi Phuong;Quynh, Doan Thi Nhu;Triet, Nguyen Anh;Long, Nguyen Thanh
    • Kyungpook Mathematical Journal
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    • v.59 no.4
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    • pp.735-769
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    • 2019
  • In this paper, we consider the Robin-Dirichlet problem for a nonlinear wave equation of Kirchhoff-Carrier type with a viscoelastic term. Using the Faedo-Galerkin method and the linearization method for nonlinear terms, the existence and uniqueness of a weak solution are proved. An asymptotic expansion of high order in a small parameter of a weak solution is also discussed.

ENERGY DECAY FOR A VISCOELASTIC EQUATION WITH BALAKRISHNAN-TAYLOR DAMPING INVOLVING INFINITE MEMORY AND NONLINEAR TIME-VARYING DELAY TERMS IN DYNAMICAL BOUNDARY

  • Soufiane Benkouider;Abita Rahmoune
    • Communications of the Korean Mathematical Society
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    • v.38 no.3
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    • pp.943-966
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    • 2023
  • In this paper, we study the initial-boundary value problem for viscoelastic wave equations of Kirchhoff type with Balakrishnan-Taylor damping terms in the presence of the infinite memory and external time-varying delay. For a certain class of relaxation functions and certain initial data, we prove that the decay rate of the solution energy is similar to that of relaxation function which is not necessarily of exponential or polynomial type. Also, we show another stability with g satisfying some general growth at infinity.