|
This article is cited in 7 scientific papers (total in 7 papers)
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.
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
Linking options:
https://www.mathnet.ru/eng/mzm11749https://doi.org/10.4213/mzm11749 https://www.mathnet.ru/eng/mzm/v106/i5/p761
|
Statistics & downloads: |
Abstract page: | 252 | Full-text PDF : | 24 | References: | 28 | First page: | 10 |
|