Geodesic
Asymptotic solutions of a parabolic equation near singular points of $A$ and $B$ types
Ural mathematical journal, Tome 5 (2019) no. 1, pp. 101-108 Cet article a éte moissonné depuis la source Math-Net.Ru

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The Cauchy problem for a quasi-linear parabolic equation with a small parameter multiplying a higher derivative is considered in two cases when the solution of the limit problem has a point of gradient catastrophe. Asymptotic solutions are found by using the Cole-Hopf transform. The integrals determining the asymptotic solutions correspond to the Lagrange singularities of type $A$ and the boundary singularities of type $B$. The behavior of the asymptotic solutions is described in terms of the weighted Sobolev spaces.
Keywords: quasi-linear parabolic equation, singular points, asymptotic solutions, Whitney fold singularity, Il’in’s universal solution, weighted Sobolev spaces.
Mots-clés : Cole-Hopf transform 
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Sergey V. Zakharov. Asymptotic solutions of a parabolic equation near singular points of $A$ and $B$ types. Ural mathematical journal, Tome 5 (2019) no. 1, pp. 101-108. http://geodesic.mathdoc.fr/item/UMJ_2019_5_1_a9/

[1] Arnold V. I., “Singularities of Caustics and Wave Fronts”, Ser. Math. Appl., 62 (1990), Springer, Netherlands | DOI | MR

[2] Khesin B., Misiołek G., “Shock waves for the Burgers equation and curvatures of diffeomorphism groups”, Proc. Steklov Inst. Math., 259 (2007), 73–81 | DOI | MR | Zbl

[3] Il'in A. M., Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, v. 102, Am. Math. Soc., 1992, 281 pp. | MR | Zbl

[4] Zakharov S. V., “Singularities of $A$ and $B$ types in asymptotic analysis of solutions of a parabolic equation”, Funct. Anal. Appl., 49:4 (2015), 307–310 | DOI | MR | Zbl

[5] Il'in A. M., “The Cauchy problem for a quasilinear parabolic equation with a small parameter”, Dokl. Akad. Nauk SSSR, 283:3 (1985), 530–534 | MR | Zbl

[6] Arnol'd V. I., “Normal forms for functions near degenerate critical points, the Weyl groups of $A_k$, $D_k$, $E_k$ and Lagrangian singularities”, Funct. Anal. Appl., 6:4 (1972), 254–272 | DOI | MR

[7] Zakharov S. V., “Asymptotic solution of the multidimensional Burgers equation near a singularity”, Theoret. and Math. Phys., 196:1 (2018), 976–982 | DOI | MR | Zbl

[8] Erdélyi A., Asymptotic Expansions, Dover Publ., New York, 1956, 108 pp. | MR | Zbl

[9] Il'in A. M., Zakharov S. V., “From weak discontinuity to gradient catastrophe”, Sb. Math., 192:10 (2001), 1417–1433 | DOI | MR | Zbl

[10] Zakharov S. V., “Asymptotic solution of a Cauchy problem in a neighbourhood of a gradient catastrophe”, Sb. Math., 197:6 (2006), 835–851 | DOI | MR | Zbl

[11] Arnol'd V. I., “Critical points of functions on a manifold with boundary, the simple Lie groups $B_k$, $C_k$, and $F_4$ and singularities of evolutes”, Russian Math. Surveys, 33:5 (1978), 99–116 | DOI | MR | Zbl