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
$$ u_{tt}-M(\|\nabla u\|^2_2)\Delta u +\int_0^t h(t-s)\Delta u(s)\,ds+a|u_t|^{m-2}u_t=|u|^{p-2}u $$
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.