Geodesic
Convergence of a family of solutions to a Fujita-type equation in domains with cavities
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 11, pp. 1902-1930

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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. 
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     author = {S. V. Pikulin},
     title = {Convergence of a family of solutions to a {Fujita-type} equation in domains with cavities},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1902--1930},
     publisher = {mathdoc},
     volume = {56},
     number = {11},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_11_a5/}
}
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S. V. Pikulin. Convergence of a family of solutions to a Fujita-type equation in domains with cavities. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 11, pp. 1902-1930. http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_11_a5/