In:
Matematicheskie Zametki, Steklov Mathematical Institute, Vol. 106, No. 5 ( 2019), p. 761-783
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.
Type of Medium:
Online Resource
ISSN:
0025-567X
,
2305-2880
Language:
Russian
Publisher:
Steklov Mathematical Institute
Publication Date:
2019
detail.hit.zdb_id:
2550622-5