Geodesic
Asymptotics of the Solution of the Bisingular Problem for a System of Linear Parabolic Equations.~II
Modelirovanie i analiz informacionnyh sistem, Tome 20 (2013) no. 2, pp. 121-128.

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Suppose we are given a bisingular initial boundary-value problem for a system of parabolic equations that contains a small parameter $\varepsilon^2$ at the second derivative and $\sqrt{\varepsilon}$ at the first derivative with respect to the spatial variable. We prove an asymptotics of any order for the solution of the problem with respect to the small parameter, without using the joining of asymptotic expansions. To this end, we apply an asymptotic method of differential inequalities. The essence of the method is to use the formal asymptotics (given in the previous paper) for constructing lower and upper solutions of the problem. By modifying the last terms of order $\varepsilon^{n/2}$ in the partial sum of the formal asymptotics, we construct the lower and the upper solutions, between which the exact solution of the problem lies.
Mots-clés : singular perturbations
Keywords: bisingular problems, asymptotic expansions. 
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M. V. Butuzova. Asymptotics of the Solution of the Bisingular Problem for a System of Linear Parabolic Equations.~II. Modelirovanie i analiz informacionnyh sistem, Tome 20 (2013) no. 2, pp. 121-128. http://geodesic.mathdoc.fr/item/MAIS_2013_20_2_a8/

[1] M. V. Butuzova, “Asymptotics of the Solution of Bisingular Problem for a System of Linear Parabolic Equations. I”, MAIS, 20:1 (2013), 5–17

[2] C. V. Pao, Nonlinear parabolic and elliptic equations, Plenum Press, New York–London, 1992 | MR