Geodesic
Dynamic Properties of a Nonlinear Viscoelastic Kirchhoff-Type Equation with Acoustic Control Boundary Conditions.~I
Matematičeskie zametki, Tome 106 (2019) no. 5, pp. 761-783

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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. 
@article{MZM_2019_106_5_a9,
     author = {Fushan Li and Shuai Xi},
     title = {Dynamic {Properties} of a {Nonlinear} {Viscoelastic} {Kirchhoff-Type} {Equation} with {Acoustic} {Control} {Boundary} {Conditions.~I}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {761--783},
     publisher = {mathdoc},
     volume = {106},
     number = {5},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2019_106_5_a9/}
}
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Fushan Li; Shuai Xi. Dynamic Properties of a Nonlinear Viscoelastic Kirchhoff-Type Equation with Acoustic Control Boundary Conditions.~I. Matematičeskie zametki, Tome 106 (2019) no. 5, pp. 761-783. http://geodesic.mathdoc.fr/item/MZM_2019_106_5_a9/