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Zh. Vychisl. Mat. Mat. Fiz., 2008, Volume 48, Number 8, Pages 1458–1487 (Mi zvmmf128)  

This article is cited in 4 scientific papers (total in 4 papers)

Convergence in the form of a solution to the Cauchy problem for a quasilinear parabolic equation with a monotone initial condition to a system of waves

A. V. Gasnikov

Moscow Institute of Physics and Technology (MFTI, State University), per. Institutskii 9, Dolgoprudnyi, Moscow oblast, 141700, Russia

Abstract: The time asymptotic behavior of a solution to the initial Cauchy problem for a quasilinear parabolic equation is investigated. Such equations arise, for example, in traffic flow modeling. The main result of this paper is the proof of the previously formulated conjecture that, if a monotone initial function has limits at plus and minus infinity, then the solution to the Cauchy problem converges in form to a system of traveling and rarefaction waves; furthermore, the phase shifts of the traveling waves may depend on time. It is pointed out that the monotonicity condition can be replaced with the boundedness condition.

Key words: conservation law with nonlinear divergent viscosity, convergence in form, traveling wave, rarefaction wave, system of waves, Cauchy problem for a quasilinear parabolic equation.

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English version:
Computational Mathematics and Mathematical Physics, 2008, 48:8, 1376–1405

Bibliographic databases:

UDC: 519.633
Received: 25.04.2007
Revised: 10.12.2007

Citation: A. V. Gasnikov, “Convergence in the form of a solution to the Cauchy problem for a quasilinear parabolic equation with a monotone initial condition to a system of waves”, Zh. Vychisl. Mat. Mat. Fiz., 48:8 (2008), 1458–1487; Comput. Math. Math. Phys., 48:8 (2008), 1376–1405

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Gasnikov A.V., “On the intermediate asymptotic of the solution to the Cauchy problem for a quasilinear equation of parabolic type with a monotone initial condition”, J. Comput. Systems Sci. Internat., 47:3 (2008), 475–484  crossref  mathscinet  zmath  isi  elib  scopus
    2. A. V. Gasnikov, “Time-asymptotic behaviour of a solution of the Cauchy initial-value problem for a conservation law with non-linear divergent viscosity”, Izv. Math., 73:6 (2009), 1111–1148  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. A. V. Gasnikov, “On the velocity of separation between two successive traveling waves in the asymptotics of the solution to the Cauchy problem for a Burgers-type equation”, Comput. Math. Math. Phys., 52:6 (2012), 937–939  mathnet  crossref  mathscinet  adsnasa  isi  elib  elib
    4. Henkin G.M., “Burgers Type Equations, Gelfand's Problem and Schumpeterian Dynamics”, J. Fixed Point Theory Appl., 11:2 (2012), 199–223  crossref  mathscinet  zmath  isi  elib  scopus
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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