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Mat. Zametki, 2019, Volume 106, Issue 5, Pages 761–783 (Mi mz11749)  

Dynamic Properties of a Nonlinear Viscoelastic Kirchhoff-Type Equation with Acoustic Control Boundary Conditions. I

Fushan Lia, Shuai Xiab

a Qufu Normal University
b Shandong University of Science and Technology

Abstract: In this paper, we consider the nonlinear viscoelastic Kirchhoff-type equation
$$ u_{tt}-M(\|\nabla u\|^2_2)\Delta u +\int_0^t h(t-s)\Delta u(s) ds+a|u_t|^{m-2}u_t=|u|^{p-2}u $$
with initial conditions and acoustic boundary conditions. We show that, depending on the properties of convolution kernel $h$ at infinity, the energy of the solution decays exponentially or polynomially as $t\to +\infty$. Our approach is based on integral inequalities and multiplier techniques. Instead of using a Lyapunov-type technique for some perturbed energy, we concentrate on the original energy, showing that it satisfies a nonlinear integral inequality which, in turn, yields the final decay estimate.

Keywords: Kirchhoff-type equation, acoustic boundary condition, original energy, energy decay.

Funding Agency Grant Number
Shandong Province ZR2019MA067
This work was supported by the Natural Science Foundation of Shandong Province (ZR2019MA067).

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/mzm11749

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English version:
Mathematical Notes, 2019, 106:5, 815–833

Bibliographic databases:

UDC: 517.9
Received: 15.07.2017
Revised: 18.03.2018

Citation: Fushan Li, Shuai Xi, “Dynamic Properties of a Nonlinear Viscoelastic Kirchhoff-Type Equation with Acoustic Control Boundary Conditions. I”, Mat. Zametki, 106:5 (2019), 761–783; Math. Notes, 106:5 (2019), 815–833

Citation in format AMSBIB
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\by Fushan~Li, Shuai~Xi
\paper Dynamic Properties of a Nonlinear Viscoelastic Kirchhoff-Type Equation with Acoustic Control Boundary Conditions.~I
\jour Mat. Zametki
\yr 2019
\vol 106
\issue 5
\pages 761--783
\mathnet{http://mi.mathnet.ru/mz11749}
\crossref{https://doi.org/10.4213/mzm11749}
\transl
\jour Math. Notes
\yr 2019
\vol 106
\issue 5
\pages 815--833
\crossref{https://doi.org/10.1134/S0001434619110142}
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\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85077028778}


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  • https://doi.org/10.4213/mzm11749
  • http://mi.mathnet.ru/eng/mz/v106/i5/p761

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