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

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
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
Related articles on Google Scholar: Russian articles, English articles

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
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
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
4. Henkin G.M., “Burgers Type Equations, Gelfand's Problem and Schumpeterian Dynamics”, J. Fixed Point Theory Appl., 11:2 (2012), 199–223
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