The initial-boundary value problem for a singularly perturbed parabolic equation in the case of double and triple root of the degenerate equation

V. F. Butuzov ; A. I. Bychkov

Čebyševskij sbornik, Tome 16 (2015) no. 4, pp. 41-76.

Résumé

The article considers two initial-boundary value problems for a singularly perturbed parabolic equation $$ \varepsilon^{2}\left(u_{t}-\Delta u\right)=f(u,x,y,t,\varepsilon), \left(x,y,t\right)\in g\times\left(0\leq T\right), $$ where $\varepsilon$ is a small positive parameter, $\Delta$ is the Laplas operator, for cases where degenerate equation $f(u,x,y,t,0)=0$ has root $u=\varphi(x,y,t)$ of multiplicity 2 or 3. Conditions are determined in each case for the problem to have a solution of boundary-layer type. Asymptotics are found and justified for epsilon $\varepsilon\longrightarrow 0$, which consist of a regular part and a few boundary-layer series. Unlike a well-known case of a single root of degenerate equation, boundary-layer solution asymptotic in case of a multiple root is constructed as a series in fractional powers of the small parameter, and these powers as well as scales of boundary layer variables depend on a multiplicity of the degenerate equation root. Another substantial difference is that three-zones of the boundary layer in a neighborhood of the initial time exist. These zones differ in decay behavior of boundary-level functions and scales of boundary-level value. The existing algorithm for constructing boundary-layer functions in case of a single root gives unacceptable results and needs significant modifications. This holds for boundary-layer functions describing boundary-layer behavior of the solution in a neighborhood of initial time as well as for angular boundary-layer functions, which play role in a neighborhood of the curve $\partial g\times(t=0)$. Proposed here modified algorithm allows for constructing unified boundary-level functions for all three boundary-layer zones. This is an advantage of this method in comparison to the method of coordination of asymptotic decomposition, where a separate asymptotic is found for each zone and then they are coordinated with each other. Asymptotic justification (i.e. existence of the solution with this asymptotic) is made using the asymptotic method of differential inequalities, that is lower and higher solutions of the problem are found using formal asymptotics. Bibliography: 17 titles.

Détail

Keywords: singularly perturbed parabolic equations, boundary layer asymptotics, double and triple root of the degenerate equation.

Bibliographie

[1] Vasil'eva A. B., Butuzov V. F., Asymptotic Methods in the Singular Perturbation Theory, Vyssh. Shkola, M., 1990 (In Russian) | MR

[2] Butuzov V. F., Nesterov A. V., “On a singularly perturbed parabolic equation”, Vestnik Moskovskogo Universiteta, 1973, no. 2, 49–56 (In Russian)

[3] Vasil'eva A. B., Butuzov V. F., “Singularly perturbed differential equations of parabolic type”, Asymptotic Analysis II, Lecture Notes in Mathematics, 985, Springer-Verlag, 1983, 38–75 | DOI | MR

[4] Butuzov V. F., Bychkov A. I., “Asymtotics of the Solution of an Initial-Boundary Value Problem for a Singularly Perturbed Parabolic Equation in the Case of Double Root of the Degenerate Equation”, Differential Equations, 49 (2013), 1–13 | DOI | MR | MR | Zbl

[5] Il'in A. M., Matching of asymptotic expansions of solutions of boundary value problems, Nauka, M., 1989 (In Russian)

[6] Nefedov N. N., “The method of differential inequalities for some classes of nonlinear singularly perturbed problems in partial derivatives”, Differential Equations, 31:4 (1995), 719–722 (In Russian) | MR | Zbl

[7] Pao C. V., Nonlinear parabolic and elliptic equations, Plenum Press, New York, 1992 | MR | Zbl

[8] Vasil'eva A. B., “The asymptotic behavior of solutions of some problems for nonlinear ordinary differential equations with a small parameter at the highest derivative”, Russian Mathematical Surveys, 18:3 (1963), 15–86 (In Russian) | MR | Zbl

[9] Butuzov V. F., “The asymptotic solution of the equation $\mu^{2}\Delta u-k^{2}(x, y)u=$ $=f(x,y)$ in a rectangular area”, Differential Equations, 9:9 (1973), 1654–1660 (In Russian) | MR | Zbl

[10] Butuzov V. F., “The angular boundary layer in singularly perturbed problems with partial derivatives”, Differential Equations, 15:10 (1979), 1848–1862 (In Russian) | MR | Zbl

[11] Denisov I. V., “Corner boundary layer in nonlinear singularly perturbed elliptic problems”, Computational Mathematics and Mathematical Physics, 48:1 (2008), 62–79 (In Russian) | MR

[12] Butuzov V. F., “On Periodic Solutions to Singularly Perturbed Parabolic Problems in the Case of Multiple Roots of the Degenerate Equation”, Computational Mathematics and Mathematical Physics, 51:1 (2011), 44–55 | DOI | MR | Zbl

[13] Butuzov V. F., “On Special Properties of the Boundary Layer in Singularly Perturbed Problems with Multiple Root of the Degenerate Equation”, Mathematical Notes, 94:1 (2013), 68–80 | DOI | MR | Zbl

[14] Butuzov V. F., Nefedov N. N., Recke L., Schnieder K. R., “On a singularly perturbed initial value problem in the case of a double root of the degenerate equation”, Nonlinear Analysis, 83 (2013), 1–11 | DOI | MR | Zbl

[15] Beloshapko V. A., Butuzov V. F., “A Singularly Perturbed Elliptic Problem in the Case of a Multiple Root of the Degenerate Equation”, Differential Equations, 53:8 (2013), 65–75 | MR

[16] Butuzov V. F., “Asymptotics of the Solution of a System of Singularly Perturbed Equations in the Case of a Multiple Root of the Degenerate Equation”, Differential Equations, 50:2 (2014), 175–186 | DOI | DOI | MR | Zbl

[17] Butuzov V. F., “A singularly perturbed boundary value problem with a multi-valued internal transition layer”, Modeling and Analysis of Information Systems, 22:1 (2015), 5–22 (In Russian)