### ASYMPTOTIC SOLUTIONS OF A PARABOLIC EQUATION NEAR SINGULAR POINTS OF \(A\) AND \(B\) TYPES

#### Abstract

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.

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- Arnold V.I.
*Singularities of Caustics and Wave Fronts.*Ser. Math. Appl., Vol. 62. Netherlands: Springer, 1990. DOI: 10.1007/978-94-011-3330-2 - Khesin B., Misio lek G. Shock waves for the Burgers equation and curvatures of diffeomorphism groups.
*Proc. Steklov Inst. Math.*, 2007. Vol. 259. P. 73–81. DOI: 10.1134/S0081543807040062 - Il’in A.M.
*Matching of Asymptotic Expansions of Solutions of Boundary Value Problems.*Transl. Math. Monogr., Vol. 102. Am. Math. Soc., 1992. 281 p. - Zakharov S.V. Singularities of A and B types in asymptotic analysis of solutions of a parabolic equation.
*Funct. Anal. Appl.*, 2015. Vol. 49, No. 4. P. 307–310. DOI: 10.1007/s10688-015-0120-1 - Il’in A.M. The Cauchy problem for a quasilinear parabolic equation with a small parameter.
*Dokl. Akad. Nauk SSSR*, 1985. Vol. 283, No. 3. P. 530–534. - 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.*, 1972. Vol. 6, No. 4. P. 254–272. DOI: 10.1007/BF01077644 - Zakharov S.V. Asymptotic solution of the multidimensional Burgers equation near a singularity.
*Theoret. and Math. Phys.*, 2018. Vol. 196, No. 1. P. 976–982. DOI: 10.1134/S0040577918070048 - Erdélyi A.
*Asymptotic Expansions*. New York: Dover Publ., 1956. 108 p. - Il’in A.M., Zakharov S.V. From weak discontinuity to gradient catastrophe.
*Sb. Math.*, 2001. Vol. 192, No. 10. P. 1417–1433. DOI: 10.1070/SM2001v192n10ABEH000599 - Zakharov S.V. Asymptotic solution of a Cauchy problem in a neighbourhood of a gradient catastrophe. Sb. Math., 2006. Vol. 197, No. 6. P. 835–851. DOI: 10.1070/SM2006v197n06ABEH003780
- 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*, 1978. Vol. 33, No. 5. P. 99–116. DOI: 10.1070/RM1978v033n05ABEH002515

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