# 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.
Published

25.02.2006

How to Cite

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 58, no. 2, Feb. 2006, pp. 236–249, http://umj.imath.kiev.ua/index.php/umj/article/view/3449.

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Section

Research articles