Two-parameter asymptotics in a bisingular Cauchy problem for a parabolic equation
type: Item
instance of: scholarly article
title: Two-parameter asymptotics in a bisingular Cauchy problem for a parabolic equation
zbMATH Open document ID: 1401.35150
DOI: 10.1134/S0081543818050164
author: S. V. Zakharov
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2018-10-24T00:00:00Z
review text: The author considers the Cauchy problem for the quasilinear parabolic equation \[ \begin{cases} \frac{\partial u}{\partial t}+\frac{\partial \varphi(u)}{\partial x}=\varepsilon \frac{\partial^2u}{\partial x^2}, & t\geq 0,\quad \varepsilon >0\\ u(x,0,\varepsilon,\rho)= \nu(x\rho^{-1}), & x\in {\mathbb R},\quad \rho>0 \end{cases} \] where \(\varphi \in C^\infty(\mathbb R)\) wiith \(\varphi''>0\) and \(\nu\) is bounded and infinitely differentiable. It is known that there exists a unique bounded infinitely differentiable in \(x\) and \(t\) solution \(u(x,t,\varepsilon,\rho)\) and the paper deals with its asymptotic analysis as \(\varepsilon\to 0\) and \(\rho\to 0.\) The author constructs asymptotic solutions in the form of series in powers of \(\varepsilon\) and \(\rho\) far from the set of discontinuity, and in powers of \(\rho /\varepsilon\) in a small domain of \(x=0,t=0\). The coefficients of this expansion are obtained from a recursive chain of initial value problems.
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 35C20
Mathematics Subject Classification ID: 76M45
Mathematics Subject Classification ID: 76D33
zbMATH DE Number: 6963752
zbMATH Keywords: parabolic equation
zbMATH Keywords: Cauchy problem
zbMATH Keywords: asymptotic solutions
zbMATH Keywords: series.
reviewed by: Lubomira Softova
MaRDI profile type: Publication
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full work available at URL: s0081543818050164
OpenAlex ID: W2887122346
description: scientific article; zbMATH DE number 6963752