There is studied the Hölder space solution of the problem for parabolic equation with the time derivative in the boundary condition, where is a small parameter. The unique solvability of the perturbed problem and estimates of it’s solution are obtained. The convergence of as to the solution of the unperturbed problem is proved. Boundary layer is not appeared.
B.V.Bazaliy, Stefan problem, Doklady AS USSR. Ser. A.11 (1986), 3–7.
2.
G.I.Bizhanova, On the solvability of the Amann problem in the Hölder spaces, J. of Math. Sciences127(2) (2005), 1828–1848. doi:10.1007/s10958-005-0145-8.
3.
G.I.Bizhanova, On the solutions of the linear free boundary problems of Stefan type, II, Mat. Zh., Almaty12(2) (2012), 70–86.
4.
G.I.Bizhanova, Solvability of the nonregular problem for a parabolic equation with a time derivative in the boundary condition, Mat. Zh., Almaty17(3) (2017), 62–70.
5.
G.I.Bizhanova, Convergence in the Holder space of the solutions to problems for parabolic equations with two small parameters in boundary condition, J. of Math. Sciences236(4) (2019), 379–398. doi:10.1007/s10958-018-4119-z.
6.
G.I.Bizhanova, Solution of nonregular multidimensional two-phase problem for parabolic equations with time derivative in conjugation condition, Kazakh Math. J.19(2) (2019), 31–48.
7.
G.I.Bizhanova and M.N.Shaimardanova, Solution of the nonregular problem for the heat equation with the time derivative in the boundary condition, Mat. Zh., Almaty16(1) (2016), 35–57.
8.
G.I.Bizhanova and V.A.Solonnikov, On the solvability of an initial boundary-value problem for the parabolic equation with a time derivative in the boundary condition, St. Petersburg Math. J.5(1) (1994), 97–124.
9.
V.F.Butuzov, On periodic solutions to singularly perturbed parabolic problems in the case of multiple roots of the degenerate equation, Comput. Math. and Math. Phys.51(1) (2011), 40–50. doi:10.1134/S0965542511010064.
10.
V.F.Butuzov, On asymptotics for the solution of a singularly perturbed parabolic problem with a multizone internal transition layer, Comput. Math. and Math. Phys.58(6) (2018), 925–949. doi:10.1134/S0965542518060040.
11.
V.F.Butuzov, Asymptotic behaviour of a boundary layer solution to a stationary partly dissipative system with a multiple root of the degenerate equation, Sbornik Math.210(11) (2019), 1581–1608. doi:10.1070/SM9149.
12.
H.S.Carslaw and J.C.Jaeger, Conductions of Heat in Solid, 2nd edn, Oxford, 1959.
13.
M.Chipot, On some anisotropic singular perturbation problems, Asymptotic Analysis55(3–4) (2007), 125–144.
14.
M.Chipot, Asymptotic Issues for Some Partial Differential Equations, Imperial College Press, 2016.
15.
M.Chipot and F.J.Correa, Boundary layer solutions to functional elliptic equations, Bulletin of the Brazilian Mathematical Society, New Series40 (2009), 381–393. doi:10.1007/s00574-009-0017-9.
16.
M.Chipot, S.Guesmia and A.Sengouga, Singular perturbations of some nonlinear problems, Journal of Mathematical Sciences, New York176(6) (2011), 828–843. doi:10.1007/s10958-011-0439-y.
17.
A.R.Danilin, Asymptotics of a solution to a problem of optimal boundary control with two small cosubordinate parameters, Trudy Inst. Mat. i Mekh. UrO RAN26(1) (2020), 102–111. doi:10.21538/0134-4889-2020-26-1-102-111.
18.
G–M.Gie, M.Hamouda, C.Y.Jung and R.Temam, Singular Perturbations and Boundary Layers, Applied Mathematical Sciences, Springer, 2018.
19.
G.-M.Gie, Ch.-Y.Jung and R.Temam, Recent progresses in boundary layer theory, Discrete and Continuous Dynamical Systems36(5) (2016), 2521–2583. doi:10.3934/dcds.2016.36.2521.
20.
E.I.Hanzawa, Classical solutions of the Stefan problem, Tohoku Math. J.33(3) (1981), 297–335. doi:10.2748/tmj/1178229399.
21.
M.I.Imanaliyev, in: Asymptotic Methods in the Theory of Singularly Perturbed Integro-Differential Systems, Ilim, Frunze, 1972.
22.
Ch.-Y.Jung, E.Park and R.Temam, Boundary layer analysis of nonlinear reaction–diffusion equations in a smooth domain, Advances in Nonlinear Analysis6(3) (2017), 277–300. doi:10.1515/anona-2015-0148.
23.
Ch.-Y.Jung and R.Temam, Boundary layer theory for convection–diffusion equations in a circle, Uspekhi Mat. Nauk69(3) (2014), 43–86. doi:10.4213/rm9584.
O.A.Ladyzhenskaja, V.A.Solonnikov and N.N.Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, “Nauka”, Moscow, 1967.
26.
A.M.Meirmanov, The Stefan Problem, Walter de Gruyter & Co., Berlin, 1992.
27.
A.V.Nesterov, The asymptotic form of the solution of an elliptic equation with the singularly perturbed boundary condition, Comput. Math. and Math. Phys.34(11) (1994), 1477–1481.
28.
O.A.Oleinik, On equations of elliptic type with a small parameter at the highest derivatives, Mat. Sbornik31(1) (1952), 104–117.
29.
E.V.Radkevich, On the solvability of the general nonstationary free boundary problems, in: Some Applications of Functional Analysis to the Problems of Mathematical Physics, Novosibirsk, 1986, pp. 85–111.
30.
V.A.Solonnikov, On an estimate of the maximum of a derivative modulus for a solution of a uniform parabolic initial boundary–value problem, 1977, LOMI, Preprint No. P-2-77.
31.
R.Temam and X.Wang, Asymptotic analysis of Oseen type equations in a channel at small viscosity, Indiana University Mathematics Journal45(3) (1996), 863–916. doi:10.1512/iumj.1996.45.1290.
32.
S.I.Temirbulatov, The problem of heat conduction with the time derivative in the boundary condition, Differentcial’nye uravneniya19(4) (1983), 85–111.
33.
A.N.Tikhonov, Systems of differential equations containing a small parameter multiplying the derivative, Mat. sbornik31 (1952), 575–586.
34.
A.B.Vasil’eva, On certain alternating boundary–layer solutions to singularly perturbed parabolic equations, Comput. Math. and Math. Phys.48(4) (2008), 618–626. doi:10.1134/S096554250804009X.
35.
M.I.Vishik and L.A.Lyusternik, Regular degeneration and boundary layer for linear differential equations with small parameter, Uspekhi Mat. Nauk12(5) (1957), 3–122.
