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

Abstract

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