Convergence of a family of solutions to a Fujita-type equation in domains with cavities

The Dirichlet problem for a Fujita-type equation, i.e., a second-order quasilinear uniformly elliptic equation is considered in domains $\Omega_\varepsilon$ with spherical or cylindrical cavities of characteristic size $\varepsilon$. The form of the function in the condition on the cavities' boundaries depends on $\varepsilon$. For $\varepsilon$ tending to zero and the number of cavities increasing simultaneously, sufficient conditions are established for the convergence of the family of solutions $\{u_\varepsilon(x)\}$ of this problem to the solution $u(x)$ of a similar problem in the domain $\Omega$ with no cavities with the same boundary conditions imposed on the common part of the boundaries $\partial\Omega$ and $\partial\Omega_\varepsilon$. Convergence rate estimates are given.