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

In this paper, we consider the nonlinear viscoelastic Kirchhoff-type equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${u_{tt}} - M(||\nabla u||_2^2)\Delta u + \int_0^t {h(t - s)\Delta u(s)ds + a|{u_t}{|^{m - 2}}{u_t} = |u{|^{p - 2}}u} $$\end{document} 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.