Singular Points and Asymptotics in the Singular Cauchy Problem for the Parabolic Equation with a Small Parameter

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.

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Correspondence to S. V. Zakharov.

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Translated by A. Klimontovich

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Zakharov, S.V. Singular Points and Asymptotics in the Singular Cauchy Problem for the Parabolic Equation with a Small Parameter. Comput. Math. and Math. Phys. 60, 821–832 (2020). https://doi.org/10.1134/S0965542520050164

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Keywords:

  • 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