Dynamic Properties of a Nonlinear Viscoelastic Kirchhoff-Type Equation with Acoustic Control Boundary Conditions. I
- Authors: Li F.1, Xi S.1
-
Affiliations:
- Qufu Normal University
- Issue: Vol 106, No 5-6 (2019)
- Pages: 814-832
- Section: Article
- URL: https://journals.rcsi.science/0001-4346/article/view/151870
- DOI: https://doi.org/10.1134/S0001434619110142
- ID: 151870
Cite item
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 → + ∞. 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.About the authors
Fushan Li
Qufu Normal University
Author for correspondence.
Email: fushan99@163.com
China, Qufu, 273165
Shuai Xi
Qufu Normal University
Author for correspondence.
Email: shuai_xi@sina.com
China, Qufu, 273165