Averaging of the Dirichlet problem for a special hyperbolic Kirchhoff equation

Abstract

We prove a statement on the averaging of a hyperbolic initial-boundary-value problem in which the coefficient of the Laplace operator depends on the space $L^2$-norm of the gradient of the solution. The existence of the solution of this problem was studied by Pokhozhaev. In a space domain in $ℝ^n,\; n ≥ 3$, we consider an arbitrary perforation whose asymptotic behavior in a sense of capacities is described by the Cioranesku-Murat hypothesis. The possibility of averaging is proved under the assumption of certain additional smoothness of the solutions of the limiting hyperbolic problem with a certain stationary capacitory potential.