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Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, 2020, Volume 60, Number 5, Pages 841–852
DOI: https://doi.org/10.31857/S0044466920050166
(Mi zvmmf11079)
 

This article is cited in 3 scientific papers (total in 3 papers)

Singular points and asymptotics in the singular Cauchy problem for the parabolic equation with a small parameter

S. V. Zakharov

Krasovskii Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, Yekaterinburg, 620990 Russia
Citations (3)
References:
Abstract: The results obtained by Il'in and his school concerning the asymptotic behavior of solutions to the Cauchy problem for the quasi-linear parabolic equation with a small parameter multiplying the higher order derivative in the vicinity of singular points are presented. The equation under examination is of interest because it provides a model of the propagation of nonlinear waves in dissipative continuous media, and the importance of studying solutions in the vicinity of singular points is explained, in particular, by the fact that even though the singular events take a short time, they in many respects determine the subsequent evolution of the solutions. In this paper, we examine five types of singular points the emergence of which is caused by different initial data.
Key words: quasi-linear parabolic equation, Burgers equation, small parameter, Cauchy problem, singular point, singular asymptotics, merging of shock waves, gradient catastrophe, Whitney cusp, Cole–Hopf transform, Pearcey function, universal Il’in solution, Lagrangian singularity, boundary value singularity, weak discontinuity, self-similarity, multiscale asymptotics, Poincaré and Erdelyi asymptotics, bisingular problem, renormalization, method of matching.
Received: 05.12.2018
Revised: 14.12.2019
Accepted: 14.01.2020
English version:
Computational Mathematics and Mathematical Physics, 2020, Volume 60, Issue 5, Pages 821–832
DOI: https://doi.org/10.1134/S0965542520050164
Bibliographic databases:
Document Type: Article
UDC: 517.956:517.958
Language: Russian
Citation: S. V. Zakharov, “Singular points and asymptotics in the singular Cauchy problem for the parabolic equation with a small parameter”, Zh. Vychisl. Mat. Mat. Fiz., 60:5 (2020), 841–852; Comput. Math. Math. Phys., 60:5 (2020), 821–832
Citation in format AMSBIB
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  • https://www.mathnet.ru/eng/zvmmf/v60/i5/p841
  • This publication is cited in the following 3 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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    References:14
     
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