Geodesic
Asymptotic behaviour and stability of solutions of a~singularly perturbed elliptic problem with a~triple root of the degenerate equation
Izvestiya. Mathematics , Tome 81 (2017) no. 3, pp. 481-504

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We construct and justify asymptotic expansions of solutions of a singularly perturbed elliptic problem with Dirichlet boundary conditions in the case when the corresponding degenerate equation has a triple root. In contrast to the case of a simple root, the expansion is with respect to fractional (non-integral) powers of the small parameter, the boundary-layer variables have another scaling, and the boundary layer has three zones. This gives rise to essential modifications in the algorithm for constructing the boundary functions. Solutions of the elliptic problem are stationary solutions of the corresponding parabolic problem. We prove that such a stationary solution is asymptotically stable and find its global domain of attraction.
Keywords: singularly perturbed elliptic problem, multiple root of the degenerate equation, three-zone boundary layer, stability of stationary solutions. 
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     author = {V. F. Butuzov},
     title = {Asymptotic behaviour and stability of solutions of a~singularly perturbed elliptic problem with a~triple root of the degenerate equation},
     journal = {Izvestiya. Mathematics },
     pages = {481--504},
     publisher = {mathdoc},
     volume = {81},
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     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2017_81_3_a1/}
}
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V. F. Butuzov. Asymptotic behaviour and stability of solutions of a~singularly perturbed elliptic problem with a~triple root of the degenerate equation. Izvestiya. Mathematics , Tome 81 (2017) no. 3, pp. 481-504. http://geodesic.mathdoc.fr/item/IM2_2017_81_3_a1/