Averaging of the Dirichlet problem for a special hyperbolic Kirchhoff equation
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.
English version (Springer): Ukrainian Mathematical Journal 58 (2006), no. 2, pp 263-279.
Citation Example: Sidenko N. R. Averaging of the Dirichlet problem for a special hyperbolic Kirchhoff equation // Ukr. Mat. Zh. - 2006. - 58, № 2. - pp. 236–249.