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

Fushan Li ; Shuai Xi

Matematičeskie zametki, Tome 106 (2019) no. 5, pp. 761-783.

Résumé

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.

Détail

Keywords: Kirchhoff-type equation, acoustic boundary condition, original energy, energy decay.

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