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

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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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     author = {Sergey V. Zakharov},
     title = {Asymptotic solutions of a parabolic equation near singular points of $A$ and $B$ types},
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     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UMJ_2019_5_1_a9/}
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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/